The episodes "200" and "201" of South Park were major examples. Cartman discovers the DNA tests to determine who his father was were tampered with, then finds out that his father was Jack Tenorman, whom Cartman had, much earlier in the series, killed along with his wife and made into chili to get back at his son Scott for stealing $16.12 from him. So he had killed his OWN father, made his half brother a cannibal, and Cartman is still more upset about being related to gingers than any of the other events.
You're Getting Old is another example. The last few episodes have been typical South Park humor, but then this episode comes along. It seems all fun and games, they're going to get better by the end of the episode.. It doesn't. Stan is breaking off from his friends, and his parents have divorced.
And it gets even worse when the following episode "Ass Burgers" reverts everything back to the way it was, defeating the entire purpose of the previous episode. With one noticeable difference: Stan is fully aware the status quo sucks, and is powerless to change it. Making it a real Downer Ending.
Long before all of that, "Scott Tenorman Must Die". It's pretty much his sled by now, but when the episode aired the aforementioned murder of Scott's parents was a major turning point in Cartman's characterization, as he went from being a naughty, immature, and somewhat idiotic 8-year old kid, to being a sadistic, manipulative psychopath.
The Season 5 episode "Kenny Dies", in which Kenny is dying from a disease, and rather than his death being played for laughs like normally, it's played for drama. And his death was permanent this time, a majority of Season 6 was about the boys coping with Kenny's death and trying to bring him back. He finally came back at the end of the season.