Forgive me if this was said already, but the debate going on was interesting and I wanted to chime in:
To start with, I believe what the OP was (gathering from the responses I've been reading) asking was for comedies with an actual story, including a beginning, middle, and end. Unfortunately what most here have been giving examples of were shows that merely reference past episodes and have a backstory (The Simpsons, Futurama, King of the Hill, etc.). I don't believe (and forgive me if I was reading in too deep). That is not the same thing. Instead, like some have already acknowledged, what he seems to be asking is for comedy cartoons like Avatar: The Last Airbender, with a definitive beginning and end, a story and plot twists, and that picks up from where each previous episode left off. Unfortunately there's very little of that in Western animation. That's not to say that there haven't been well-written and genuinely hilarious comedy cartoons that are episodic, but I can see where some might be coming from. Some may not be satisfied with seeing the same premise over and over again, week after week, where the status quo is almost always upheld (The Simpsons is notorious for this). It's just not the type of show for them.
I'm afraid I'm just as in the dark as the one who posted first about actual Western animation of this kind. I'd like to see more variety in these things myself. But I understand that there are reasons for this: one reason that shows like these opt not to go in that direction is because it makes them harder to syndicate (please correct me if I'm wrong on this). A show like Spongebob Squarepants can be watched in any order and understood because the main characters have fixed residences, occupations, and personalities. Whereas one cannot watch the final episodes of Avatar without knowing what has occurred before. Neither of these methods make a show 'bad' (there are continuity-based shows that are downright terrible). It just might not be that person's cup o' tea.
I hope that makes sense.