The Fantastic Four were actually a groundbreaking feat in superhero comics because of this trope. Before their creation, it was unthinkable for a superhero team to have such blatantly dysfunctional interpersonal dynamics and depressing personal issues. Not including all the crap that they go through after they get superpowers, from the very beginning the group includes a guy whose entire normal life was ruined by getting turned into a giant, hideous rock monster who can't hide his identity. And as for screwed-up backstories, they have the guy whose former friend hates him with such ablinding passion that every single thing said friend has done in his career as an evil, world-dominatingMagnificent Bastard can be traced back to his obsessive desire for revenge.
In Ultimate Fantastic Four their issues are so bad that the team eventually falls apart and Reed becomes Ultimate Doctor Doom.
And that's not even getting started on Spider-Man...
The Young Avengers, if anything, are worse. We've got everything from accidentally almost killing bullies to juvenile delinquency, steroid abuse, and rape.
This explicitly seems to be the premise of Avengers Academy. The student body consists of six teenagers who were forcibly and torturously given powers by Norman Osborn. They include a guy who's stuck in a metal body, a radioactive girl who gave her parents cancer and a girl who's slowly dissipating. They're all revealed to be "damaged goods" who are most likely to turn into villains and are given training to preempt that. Accordingly, their instructors are some of the most troubled members of the Avengers, who are supposed to benefit the kids with their experience of dealing with their problems.
All together now: Frank Castle's family was accidentally murdered in a Mafia shootout, Frank has been killing criminals ever since as punishment... for them or himself, as he was thinking of leaving his family due to his PTSD and newly-discovered Blood Knight tendencies.
Strangely enough, many of the villains get dysfunctional backstories as well, despite rarely making it to the end of the book: Nicky Cavella was manipulated into murdering his family (and repeatedly raped) by his aunt, a creepy homeless guy living in a pile of corpses had to eat his way out of his morbidly obese mother when she had a heart attack and fell on him, Barracuda's sociopathy comes from his being unable to find and kill his abusive dad, the Kingpin had his abusive dad eaten by rats and underwent Prison Rape...
On Runaways, the unifying aspect of the group is that everyone had super villain parents.
Joss Whedon's run added Klara, a twelve-year-old abused child bride from 1907.
Aside from Klara, Whedon's run dials this trope back a bit; surprising, considering he usually gleefully subjects characters to as much as they can take and then some. Quite a few of the Runaways actually come out better or no worse off from their adventure. Chase gets new weapons and manages to move on from Gert's death, resisting the temptation to go back in time to save her. Xavin manages to overcome his/her gender issues. Nico gets a new staff and powers (kind of like Willow). Molly remains unchanged. About the only one messed up further is Victor; he falls for a new girl, Nico dumps him, and new girl doesn't go with him back to the present. One gets the feeling Whedon didn't like Victor.
Transformers: More than Meets the Eye gives us the crew of the Lost Light. Almost half of the crew suffers from extreme mental, social, and personality disorders. Rodimus is a self-loathing Glory Hound, Ultra Magnus is Super OCD, Drift is implied to be suffering PTSD, Swerve is a Stepford Smiler who feels like he doesn't have any friends, Tailgate is a Phony Veteran who secretly hates himself, Cyclonus has difficulty forming emotional attachments, Trailbreaker is an alcoholic, Chromedome is constantly depressed and has suffered the Cartwright Cursefour times, and Whirl is literally insane. Even the Only Sane Man, Hoist, has a traumatic past that has left him with a crippling phobia of being alone. Just to top it off? They only have one therapist... who might have just as many issues as the rest of the crew, if not more.
Elsewhere, there's the Wreckers. Aforementioned Whirl isn't even the worst example present. There's three ways of leaving the Wreckers: Death, dishonourable discharge and dementia. There's Sandstorm, who went insane after the war and started killing people he thought had escaped justice, Guzzle, who only joined so he could murder Kup, and then went insane and started violently killing enemies, Impactor, Megatron's former drinking buddy who caused the big guy's Start of Darkness, and got thrown in prison for summarily executing all of Squadron X, and Roadbuster, who used to be a drill instructor who heard a voice telling him to push his trainees to the point of death, sometimes even giving them a nudge over the edge. And those are just the big names.
Then there's the Dinobots, or, as they were originally called, the Dynobots. Originally formed before the war, Grimlock and Slug met when they were both in the brig for pissing off theri commanding officers (Slug having shot his). After teaming up with four other misfits, Swoop, Sludge, Snarl, and Skar, they became a black ops unit that was highly effect in spite of Slug's insistence that they not be an actual team. Things were actually going well for them until an op looking for terrorist lead to an encounter with viscous cyber-morphic predators that resulted in the death of Skar and the other five being left with unstable, uncontrollable altmodes of those same creatures. When the Cybertronian government decided to use them to study Skar's dynamic altmode adaption program rather than cure them, so they went on the run, feeling betrayed. They wound up rallying to help Optimus Prime when he called for everyone to rise against the Decepticons, but by then their altmodes' aggression was starting to seriously affect them and they turned to crime as a means of earning enough money to purchase passage off Cybertron before they became a danger to innocents. They ended up receiving help from the Autobots and joined the team, but even upon being cured they still stayed vicious Blood Knights who committed numerous war crimes during the war. Against orders, they chased Shockwave to Earth, and Grimlock set up a dead-man's switch on their ship without telling the others, which caused it to fire on the battlefield as they fought Shockwave. This triggered a volcanic eruption that buried them all in lava. When they were dug out in the present day, the others confronted Grimlock over this, who defended himself by saying that yes, of course he did it because the Dynobots are a bunch of vicious amoral bastards who do that sort of thing. The others agreed that that was a valid reason. They could barely function while the war was raging and there was an actual need for a group that "does the things that allow heroes to stay heroes", once the war's over they're left without a purpose in life and unable to function in normal society.
Grimlock: We watch each other's back, we stick together through thick and thin, and no one ever tells us what to do!
And then there are the Scavengers, who were at exactly the bottom of the Decepticon hierarchy while there was still a Decepticon hierarchy. Fulcrum is a paranoid coward, Crankcase seems to have a form of anxiety, Krok suffers from trauma and is in severe denial, Misfire is just kind of nuts, Spinister is delusional, paranoid and violent, even he is a genius with his hands, Grimlock is brain-damaged, and cons4eva is a renegade Dire Wraith.
With few exceptions, almost all of the X-Men have tragic pasts, poor childhoods, dead parents or all three. This is compounded by the series' use of Expansion Pack Past, which tends to add on progressively more tragedies in the character's personal history the longer the series goes on, continually "revealed" to the audience whenever a character is focused on. Some of this is retooling to more clearly explain the animosity of mutants rejected by society, but a fair bit has existed from the team's earliest days.
X-Statix had this essentially as its core premise — this was a team of celebrities, not heroes, and as such, extremenote read "clusterfuck" personalities clashing is to be expected. But to go into detail: the Orphan has an adversarial relationship with the Anarchist, who in turn is bitter rivals with the Spike. Phat and Vivisector don't get along with anyone except occasionally each other. El Guapo disrupts the team when it starts to gel, the Mysterious Fan Boy's naivety grates on everyone but forces them to at least pretend to like each other, Venus Dee Milo draws flak from fans and the media, which lessens her stock among the team members themselves, and Dead Girl is just... weird. U-Go Girl is probably the only one who functions semi-normally within the group.
Who are The Ultimates? A soldier of WWII who suddenly found himself in a future world where nothing is like he remembers. A genius, billionaire, playboy, philanthropist; who is secretly dying of brain cancer. A new-age guru who claims to be a Norse god reincarnated (or is he?). A wifebeater with an inferiority complex. A closeted mutant who usually gets in toxic relations. A scientist broken down by his inability to crack the super soldier formula after so many years of study, and with a dominating girlfriend. Two master assasins. Two mutant brothers treated like dirtbags by everyone, including their own father. Unsurprisingly, loads of drama ensued.
Three words: The Bat Family. We might as well just say Gotham is Dysfunction Junction. Just living there practically counts as an angsty past.
Bruce Wayne's parents were shot dead in front of him when he was eight. He dealt with this by repressing his emotions, dressing as a bat to fight crime, and reguarly inducting children and teenagers into his basement cult.
Dick Grayson's parents were killed in front of him when someone sabotaged their trapeze act. He was taken in by Bruce, an emotionally distant billionaire, who would later sack him as Robin and kick him out when he became old enough to disagree with him. Bruce would then replace him with Jason, further straining their relationship. Later, after they've made up, Bruce is killed. This forces Dick to become Batman (something he's never wanted to do) and take on Damian so that the kid didn't return to the League of Assassins, damaging his relationship with Tim in the process. Bruce eventually returns, but soon after Damian dies, followed swiftly by Dick himself being unmasked, killed, and resuscitated. Whilst he's emotionally broken from the aforementioned events, Bruce kicks the ever-loving shit out of him until he agrees to go undercover at Spyral, something he really doesn't want to do because it involves telling everyone he loves that he's dead. When he returns, they're all mad at him as he predicted and the fact that Bruce beat him up until he agreed to do it is never brought up (possibly because Bruce had amnesia at that point). The fact that this guy is so well-adjusted is, frankly, a miracle.
Barbara Gordonused to be a gymnastic crime fighter until she was shot and paralyzed by The Joker to break her dad.
Jason Todd, whose parental issues include thinking himself The Unfavorite because Batman didn't avenge his death at the hands of the Joker, worded thusly: "I'm talking about killing him. Just him. And doing it... because he took me away from you." Before meeting Bruce he rarely saw his biological father, a criminal who cycled into and out of prison until his boss (Two-Face) finally offed him, and Jason had to look after his mom as she died, likely due to drug abuse. His biological mother, without any remorse whatsoever, later lured him into a trap set by the Joker and calmly looked on as he was beaten within an inch of his life. He then died, and Came Back Wrong. It all goes downhill from there.
Tim Drake, whose parents are dead (including step parents and fake ones). Said parents were also neglectful towards him. When Jason Todd, who Tim idolized, returns from the dead, he tries to kill him. Then, towards the end of his run as Robin, a frankly ridiculous number of his friends and allies died leading to him becoming a far more angsty character than he was when he started out. Damian then turns up and tried to prove himself to his father by murdering Tim, because how else would a child assassin prove themselves? Then Bruce seemingly dies, Dick takes Damian on as Robin, effectively forcing Tim out of the role. Everyone then proceeds to treat Tim like he's crazy for insisting that Bruce is alive...not that he has any evidence to support his claim given that it's essentially the last hope of a grief-stricken, depressed teenage boy.
Stephanie Brown, aka the Spoiler! Her father was Cluemaster, a third-string Batman villain who was a Riddler rip-off. So when she was growing up, her home housed a steady stream of criminals, her dad was in and out of jail, and her mom was addicted to prescription drugs. Her father used her in his various plots, once getting her kidnapped and put in danger of death as part of an elaborate trap for Batman. During high school she got pregnant and ultimately made the heartbroken decision to give up her baby for adoption. She became Robin, was fired for doing the job, and promptly accidentally started a massive gang war that killed hundreds of people and ended with her nearly dying at Black Mask's hands. When she was able to return to Gotham she found her boyfriend (Robin) still loved her but felt betrayed that she left him thinking her dead while all his family and close friends died in quick succession, and then continued to lie to him after her return so finally joined the ranks of those who told her to stop acting as Spoiler. Since she has become Batgirl and begun a relatively well-adjusted life as a college freshman while her mother (now off drugs) works a steady job at a hospital.
Damian Wayne is the son of Bruce Wayne and Talia Al Ghul. He was raised in the League of Assassins but rejected their ways in favour of working with his father and Dick Grayson as Robin. His mother responded by disowning him and, eventually, sending his clone to kill him. He got better. Being the only Batfamily member to be forced on Bruce Wayne rather than chosen, he's extremely insecure of his place. Trying to hide this with arrogance only makes him more unpopular within the family.
Some universes also include Huntress as a member, whose parents were shot dead in front of her when she was eight.
And that's without counting the myriad of traumas and psychoses behind almost all the Batman villains.
The Doom Patrol. To descend into just how screwed up everyone in that group's roster is would take up the whole page.
To give you a taste: The team roster for Keith Giffen's run includes a man whose brain was preserved by putting it in a robot, an off-kilter Energy Being with identity issues, a former B-Movie actress with low self-esteem whose ex-husband is a telepathic stalker, a girl with 64 multiple personalities, and an amoral bastard of a Mad Scientist who treats losing his legs as an inconvenience.
The Teen Titans. It's arguably more of a support group for superpowered teens than an actual team of superheroes. Considering how many of them have died and/or gone insane, it doesn't do a very good job.
Watchmen comes close; the only superheroes in it that are pretty well-adjusted are the two Nite Owls and the second Silk Spectre. They have some of their own baggage too, but they're generally pretty normal people (Nite Owl II, in particular, is ultimately a mildly depressed guy who feels most comfortable and meaningful—personally and, uh, sexually—in superhero mode, a major step up from a mainstream hero like Batman).
Jesse Custer has familial issues and then some, although not what you would expect.
His parents were fine, upstanding people who did not balk at showing their love. Sadly the same did not hold for his maternal grandmother, who in her attempts to control Jesse's upbringing kept him and his parents hostage from an early age on, and had his father murdered for attempted escape. His mother was later to suffer the same fate, for attempting to intervene when Jesse was to be punished by being left to stew in his own feces and urine, with no nutrition, in a submerged coffin, for a week.For using a swearword in anger against her personal henchmen, who had recently murdered his puppy.
Can somebody say "Todd Casil" also known as Squee? His parents explicitly state on several occasions "I wish you'd never been born," his neighbor is a homicidal psychopath, and he has a 1-sided friendship with the son of Satan!