Supergirl's Pre-Crisis origin is pretty horrible to begin with. She manages to survive the destruction of Krypton by her father sending a city into space with Kryptonian super-tech only for everyone to slowly die of radiation poisoning due to being on a big block of Kryptonite. Sent to live on planet Earth in order with her cousin, he puts her in an orphanage to keep her from being targeted by his enemies, only to have to spend years undercover so most people didn't even realize a Supergirl existed.
In imaginary story Superman vol 1 #149 Lex Luthor manages to kill Superman-. Supergirl takes him away and brings him to Kandor where he is punished. However she has lost her cousin and she and Krypto have to carry on his name.
In Superman vol 1 #282Kara explains her cousin◊ that she is thinking of giving up her Supergirl identity because she wants a normal life. The scene of a tear-jerker itself, but it's even more heart-wrenching when you remember that she would never have an opportunity to lead a normal life because of the Anti-Monitor.
Superman: Still thinking about giving up your Supergirl identity, Kara? Supergirl: I don't know, cousin Kal-El — Maybe! This life of a super-heroine takes up too much of my time... sets me apart from everybody else! I want an ordinary life — with a husband and children some day — free to do what I choose!
In Action Comics #317, Lena thinks her friend Linda has betrayed her and wants to move out of the country because Linda was her only friend and now she has no one and no reason to keep living in Midvale.
The Death of Supergirl in Crisis on Infinite Earths issue #7 remains one of the most moving storylines in comics until that point. Supergirl, Superman's last remaining living family, willingly gives up her life to save the Multiverse.
Zor-El and his wife Alura In-Ze cradling their child's body in Kandor.
Kara: We don't do it for the glory. We don't do it for the recognition... We do it because it needs to be done. Because if we don't, no one else will. And we do it even if no one knows what we've done. Even if no one knows we exist. Even if no one remembers we ever existed.
In DC Retro-Active Superman 1980's, Earth-1 Superman has a nightmare where -among other things- he sees Supergirl dying. When he wakes up he barely remembers his dream, but he is very shaken and frightened, and he calls his cousin to ask her to be careful. Kara dismisses his fears... and then she mentions that the morning sky is red...◊
At the very end of Supergirl's second solo book, Dick Malverne runs into Linda and kisses her. Twenty years later, Solo #1 tied up that loose end: Dick sought Linda out because he was dying from cancer. On his death bed, he confessed he always loved her and knew she was Supergirl. They gave each other one last kiss goodbye, and he died later that night. Summarized here:
That was the last time I saw Richard. Tonight I know what it is to be human. Tonight my super skin is still invulnerable, but my Richard is never coming back, and my heart is broken into a million pieces.
The Matrix Supergirl discovers that the Lex Luthor she fell in love with, a handsome Australian businessman who has always kept a Villain with Good Publicity front, is actually the original Lex Luthor wearing a cloned body. Worse, not only has he never cared about her but he'd been secretly trying to clone her as an army of genetic soldiers in the meantime.
Matrix has struggled with defining her humanity and whether or not she thinks she truly is a person and not some thing, and then when it looks like Buzz and Tempus Fugate have murdered Fred and Sylvia Danvers, Matrix goes completely beyond the Despair Event Horizon. And unlike when she went berserk before, she's heartbreakingly terrifying as flames burn around her and she cries tears of blood. As she confronts Tempus and Buzz, Matrix is surrounded by ghastly apparitions of Superman, the Kents, Lex, and Linda Danvers' friends and parents while thinking to herself how completely wrong she was about doing good when suddenly, an image of Linda appears before her and tries to convince Matrix she's more human than she believes. Unfortunately, Matrix is so past that point she starts screaming about the futility of it all, how people keep dying no matter how much she tried to do the right thing. Thankfully, she's snapped out of it thanks to the timely intervention of Wally the God Boy channeling through Buzz, who asks Matrix if she's really prepared to cross the line and kill Tempus.
The entire Many Happy Returns arc of Peter David's run is one long series of gut-punches, Heartwarming Moments and Moment of Awesome. The Pre-Crisis Supergirl is saved from death by Hal Jordan's version of the Specter only to slowly realize the universe is coming undone due to her being necessary to kill the Anti-Monitor. Linda Danvers tries to take Supergirl's place in the Pre-Crisis continuity, eventually marrying and having the Pre-Crisis Superman's child, only to discover it won't work. Linda Danvers isn't strong enough to hurt the Anti-Monitor enough to save the universe. Linda is separated from her child and the 15-year-old Supergirl is forced to go back in time, willingly, to save everything. The Specter promises to protect her child but uses Exact Words and Linda is separated from her offspring, possibly forever.
Supergirl's father Zor-El being killed by Reactron, days after his daughter discovers him having escaped the destruction of Argo City and fleeing with her mother to the Bottled City of Kandor.
Supergirl's discovery her friend Superwoman is an enemy and actually Lois' sister Lucy.
In Supergirl Vol. 5 issue #43, Kara writes a long letter addressed to her deceased father, telling how badly she misses him, and explaining why she didn't choose his Guild.
Supergirl: I love you, Dad. I don't think I told you that enough. I love you. And— and I miss you. Every day. And I always will. Love, Kara Zor-El, on her Guilding Day.
Supergirl losing not only her father but her mother as well as every surviving member of her species but Kal-El and the Phantom Zone criminals when Sam Lane blows up New Krypton.
As fighting alongside Supergirl, Bizarrogirl learns morality in human terms and realizes she killed a person. She chains herself to a rock and she asks if she'll ever forgive herself some day. Supergirl, who was just so shattered after New Krypton when her people raided Earth and killed thousands of innocent bystanders, understands her too well.
Supergirl and Power Girl never reconciled.
Supergirl discovering the dead remains of Argo City orbiting a blue star. She is willing to lie down and die amongst the remains of her parents but chooses, after a hallucination of her father, to fight on.