Much like The Simpsons, Family Guy realistically wouldn't qualify for Getting Crap Past the Radar since that trope is mostly for shows that are family-friendly or kid-friendly, and Family Guy is neither, despite it having a Periphery Demographic of younger viewers. The show carries a TV-14 rating in America and a 15 rating in the UK for a reason.
On the other hand, the writers do go out of their way to make FOX censors squirm and cry with Seth Mac Farlane's brand of offensive, often dark humor on this show and, to a lesser extent, his other shows (American Dad! and The Cleveland Show). FOX does catch some of this and ask MacFarlane and company to change it (with the original scenes appearing on [adult swim] and/or DVD), but people make mistakes, and sometimes they don't catch everything:
In Brian Sings and Swings, Meg inadvertently joins the Lesbian Alliance Club after making friends with a teenaged lesbian who thinks Meg is one too. She discovers this from a poster that says "See You Next Tuesday" on it.
In that vein, Stewie Griffin: The Untold Story has Quagmire setting up a "Cross-Country Tour" where he bangs a different woman in each state. Brian reads it and notices Quagmire misspelled "country" in "Quagmire's Cross-Country Tour".
Brian: 'Quagmire's Cross-Country Tour'. Uh, shouldn't there be an "O" in 'country'?
In One if By Clam, Two if By Sea, Stewie mouths "fuck you" to Brian to test his ultrasonic hearing skills after Eliza wets herself and goes back to speaking Cockney English (dogs can hear at frequencies that humans cannot, which is why a dog whistle is silent to humans, but dogs will immediately react to it).
Brian: I'm telling.
Stewie: No, wait! I said 'vacuum'!
There's a joke about a "Cleveland Steamer" in "Mr. Saturday Knight". What makes it even funnier is that the original joke used the less-vulgar term "half and half" which the censors wouldn't allow. Seth said it was one of the most vile jokes they'd ever gotten away with, and that was mostly because the censors assumed that the writers had made the term "Cleveland steamer" up and didn't know it was an actual sexual term.
The episode PTV is one big episode that got past the radar entirely (and yet, the real FCC found the episode hysterical rather than brand Seth MacFarlane as a menace to society, like the Parents Television Council does to MacFarlane on a regular basis). In fact, one song from the episode shows all the episodes that got crap past radar from that point (and some new stuff they added, like Stewie spanking Brian in S&M gear, Peter replacing Janet Jackson on the infamous Wardrobe Malfunction incident, and Peter flashing his breasts on Howard Stern's radio show). Not to mention that song actually has the word "fellatio."
Carter: Ahh, that was excellent. When I clenched it, you took your fingers away. You were right to do that.
When Babs wakes Carter in the middle of the night he says "It's okay Lois, Daddy's just taking your temperature."
In "Tom Tucker: The Man and His Dream", Chris brings home a girlfriend who looks like a younger Lois. The rest of the cast is surprised, and Stewie remarks that Chris was getting "Oedipussy". Brian even asks Stewie if they could even say that, to which Stewie replies, "We just did."
From Sibling Rivalry, a lesbian (who looks like the gym teacher who told Meg's class to take their tops off, get in the shower, and bounce around for her pleasure) and her partner tell the sperm bank clerk that they want a baby and need the following: a vial of sperm and an applicator that looks like Jodie Foster's knuckles.
In season two's "I Am Peter, Hear Me Roar", Quagmire says "Hey, look at me! I'm the man in the boat!" note "man in the boat" is a slang term for the clitoris. This episode was made pre-cancellation, when the show wasn't known for it's, uh, crude humour as much as it is now.
The cutaway from season seven's "FOX-y Lady" where Ellen Degeneres's breath smells strongly of fish.
From season eleven's "Bigfat":
"You can get your HOCKEY PUCKED in Canadian nudie bars."
From season ten's "You Can't Do That on Television, Peter":
Stewie: "Do you think Mother Maggie is married to Father Fa-oh, probably not."
Peter warning Lois not to nag him, because he "...can have any 3-year-old he wants." Especially shocking given the news of British children's show host Jimmy Savile "having any kid he wanted" and the BBC didn't know about it until Savile died.
In the episode "New Kidney in Town", where Peter takes a liking to energy drinks, he buys a cow. His teenaged son, Chris, also starts drinking energy drinks. Both of them then live in Caffeine Bullet Time. We then see Peter milking the cow so fast it causes the cow's udders to burst into flame. A few seconds later, we see Chris panicking and running downstairs, with his pants on fire.
From "Bill and Peter's Bogus Journey":
Secret Service Agent: It seems (Bill Clinton) blew a tire in front of your house.
Peter: Boy, that guy is really horny, isn't he?
"No Chris Left Behind" has Morningwood Academy.
In season ten's "Amish Guy", Quagmire reads off his food journal, which turns out to be the names of women with food names that he has "eaten" out ("Peaches, Ginger, Honey, Candy, Olive...")
In "Forget-Me-Not" Brian asks Peter if he wants to hit the Clam, Peter's response:
Peter: Nah, Lois ain't in the mood, let's just go to a bar