The episodes "200" and "201" 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 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.
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.
"Raisins": Wendy dumps Stan due to him not spending time with her. The plot is complicated by the fact that neither Wendy or Stan care to talk themselves and instead send friends to do so, leading to more poor communication getting in the way. Stan attempts to win her back, only to see that Token has started dating her. He falls into a depression and goes goth, up until some encouraging words from Butters cause him to go back to his normal lifestyle. It ends with Stan calling Wendy a bitch, and flipping off her and Token. Wendy and Stan wouldn't get back together until "The List"- a full four seasons later. To even show that Wendy's relationship wasn't reset and that things wouldn't be so easy, the animators had her and Token paired together in the background in "The Jeffersons" while "Follow That Egg!" ended in a dejected Wendy walking away upset after Stan insulted her.
Butter's sub-plot in this episode can be considered this too. Butters meets a girl named Lexi who works at the Raisins restaurant after she was nice to him (she was only doing her job, though.) Butters spends the rest of the episode obsessing over her, thinking she was his girlfriend, eventually learning the hard truth that she never liked him at all. He then sits in the rain, wallowing in his "beautiful sadness", as he puts it.
"1%": Cartman's toys wind up murdered, and the culprit winds up being Cartman himself, destroying his childhood.