Short answer.
Any and all DCAU
Just about any Comic Book Hero related Direct to DVD release (That wasn't clearly for kids)
Planet Hulk, Green Lantern First Flight, Hulk Vs., Ultimate Avengers (1 & 2), Batman: The Red Hood (Awesome), Justice League Crisis, Superman/Batman etc. etc.
Reboot (TV Series)
Beowolf (Animated film)
Pixar's Up (animated film....it got a freakin' best picture nomination in 2010...It was nominated alongside Prcious, The Hurt Locker, Avatar, District 9, The Blindside....)
Primetime Cartoon Sitcoms (Simpsons, Futurama, Family guy, etc. etc.etc, Yes I read the OP, long answer explains this)
Lucy Daughter of the Devil
LONG ANSWER!!!!
First off, "how animated films and TV shows can be as mature as live action ones". Wouldn't it be easier to show how live action shows and movies can be way dumber? (See any film starring Pauly Shore...or the Jersy Shore even).
Second I take issue with "By mature, I do not mean it includes "mature humour". As a rule of thumb, most comedy will not count due to this.". Comedic shows (animated and non) are very capable of dealing with serious/mature issues.
Now this is just my opinion (I'm not saying YOU'RE WRONG), I'm just pointing out that I feel you're trying to make an argument (for your essay) and immediately limiting yourself by exempting a considerable portion of the material available because of some belief that comedy and mature content are mutually exclusive (If you're trying to prove that Animation can be as DRAMATIC as live action, then thats a different kettle o' fish). Remember that Animation is entertainment, and entertainment (particularly now) is driven by commerce, and sure you can make a serious tv show or movie starring dramatic actors and get backing for it, the idea of a serious dramatic cartoon, unless it was attached to some sort of existing franchise...say like the Matrix (The Animatrix), or a comic book animated features attached to a major company (Marvel/DC, see Thor, Hulk, Batman, Justice league, Wonder woamn and Green Lantern Animated films that have flooded the market in recent years, to mention a few, funny how A live action Wonderwoman Movie has been in Preplanning forever, a live action TV show had a pilot made but will likely never see air, but it was no problem getting her out an animated DVD),or character that has already been well established (Spawn or Hellboy) that release direct to DVD. A show like the Daily Show (not animated I know but I'm making a point here) is clearly comedy, but it is adult oriented, to say nothing of relevant issue which it uses comedy to make its point. Also throughout the history of entertainment, comedic material has often been used to mask more serious social content.
Shows like Family guy, simpsons, futurama, and others, are in fact comedies, but are in no way made for kids (okay maybe Simpsons is a little closer to family friendly, but Family guy is NOT). I would argue that the number of serious celberties who have lended their voice and image to such shows should give them credit. Al Gore appeared as himself on Futurama...former VP, Former emporer of the moon, Environmental activist. He was getting his message out through a well established medium. I also believe he was involved on an extra feature from one of the Futurama DVDs. Futurama deals heavily with parody, but the difficulty with such a comedic approach is that you have to parody things the audience will recognize, and a fair bit of the content that they have lampooned has been stuff a younger audience isn't likely to have known.
And what of South park? A comedy that is SO Adult oriented.
Beavis and Butthead weren't for kids...mind you mature was a label it didn't really earn.
Someoen mentioned Daria, the antithesis of Beavis and Butthead, it had humor, but wasn't a straight up comedy. It was actually a pretty mature look at highschool and workings there of, with the sarcastic outcast main character not always being as right as she thought she was (But not always wrong either)
Justice League and Justice League Unlimited (Heck the Batman and Superman animated series that preceded them as well).
Honestly if you look at the story and character progression in those, there was plenty of stuff that really would have been over most kids heads (But they still could have enjoyed it). Sure kids could enjoy it, but most of the content was mature. Hell, they did a Batman Mr Freeze direct to video movie that probably came out within a year or two of the Batman and Robin Movie (With Mr Freeze) and storyline alone, I'd say the animated one was hands down 1000 times better than that Joel Shumacker garbage. (Funny how a well recieved cartoon about Batman could only spawn direct to Video/DVD content, while a live action farce got a feature film)
The 2nd season of JLU completed a season long story arc (Begun at episode 1, and carried as an ongoing backstory with a 4 part finale) with Superman ready to kill Lex Luthor, and then the final episode of JLU season 2 (About Batman Beyond, "Epilogue" I think), just watch it and then show me a 6, 8 or 10 year old that could process that!
Several years ago (early to mid 90s) a company named Mainframe made a CG Cartoon called Reboot, about the lives of characters who lived in computers. The first 2 seasons it was pretty much a generic kids cartoon (it aired on ABC), nothing too bad, good always won, evil scampered away, the occasional morality lesson. But they split off from ABC and the 3rd and "4th" seasons (season 4 was actually a collection of few 2 hour (made for TV) movies, which ended on acliffhanger they still have yet to resolve)and established more mature and complex stories. Violence, death (or death like occurences), implied sex. This got good airtime in Canada, don't know if any US networks picked it up.
If you look at the history of western animation you actually see a strange progression. Feature animated films date back to the 30s and 40s, possibly earlier (Disney, Snow White, Dumbo, others) and they weren't strictly for kids.
The Flintstones were in the 60s (I think, or 70s) and that was actually a primetime show, like the Simpsons of that time. Around that time there was also Yogi Bear and others, which were more kid oriented. Much of comedy in all of them was classic slapstick, or vaudvilliean even.
At some point it was decided that cartoons were for kids, and this belief was held for some time (in western society), and still is by some.
But with shows like the Simpsons, Family guy, and Futurama animation is getting more credibility. Hell Pixar's "Up" got a Best Picture nomination back in 2010
In fact, if the point of your essay is to prove the old image of cartoons not being a vehicle for mature stories or concepts, you should really compare some 70s cartoons like Superfriends or Batman to their more modern counterparts, like the DCAU series, or the direct to DVD features.
Its funny you said you didn't want to include any "avant garde", because that stuff is more likely to be less kid oriented, since that would be more artistic (And focused on telling a particular story or getting a point across instead of being commercially successful), while most mainstream western animated feature films target the widest possible audience (Its all about the bucks)and with the prevailing notion still there about cartoons being for kids, they (believe that they) can't afford to release something for adults that kids won't enjoy (Or worse yet, be prevented from seeing) and still be profitable.
Anyways, hopes this helps some, not trying to pile on, just looking to impart some knowledge, and a (hopefully coherent)different point of view that might provide you some ideas (if nothing else).