Teen Titans (the original) is probably a good example of drama and comedy done in a way that's more reminiscent of the themes we see from anime, but then again it came from DC Comics and they tried to tie it to Eastern animation as much as possible (art style and the opening done by j-pop sisters). It dealt with a lot of different themes, it was dark, it was mature, but it also never broke itself from being appealing to kids.
That's probably the best that Western animation has done in terms of a mature animated series. Where is the Western equivalent to Ghost in the Shell? Gunslinger Girl? Black Lagoon? Cowboy Bebop? Neon Genesis Evangelion? Where is the danger, the threat of death, the psychosis, the dynamic characters, the dark themes? I've not found anything that deals with that beyond Teen Titans, and even then it had way more comedy and was less invested in darker themes than any of the aforementioned series. Western animation is either for kids or young adults (from my observations it's rare to find anyone over the age of 30 that likes Family Guy/The Simpsons/South Park etc.), and if you want anything deeper you go for tv series, miniseries or movies. Japanese animation has a far broader scope than Western animation, so people in the West who aren't exposed to Eastern animation automatically apply all the stereotypes of Western animation to Eastern animation, and that won't change until Western culture completely assimilates Eastern animation (which it's done a pretty damn good job so far I've got to say) or something that breaks the mold in the West becomes super popular and influences further changes (since otherwise it'll just be considered a one-off special deal).