1. Quagmire
As well as basically endorsing paedophilia, Family Guy actively allow an out-and-out serial rapist to run around Quahog all day like it's the most hilarious thing in the world. "Hahaha! Quagmire just drugged that girl and now he's dragging her by her feet into his room so he can rape her." The whole point of Quagmire's character is to create controversy; from frotteurism and erotic asphyxiation to zoophilia, incest and necrophilia, there's no line this wildly-overblown character hasn't crossed. It's easy to find some of this stuff humorous, if you disengage your brain and allow Family Guy to wash over you, but then Quagmire's behaviour becomes controversy for the sake of controversy.
Of all the disgusting characters Family Guy could have killed off, they decided to kill off just about the only one with any kind of morality, leaving us with a show of morally-bankrupt, overblown, running-gags.