You often hear animation writers complain that there was no time to make a story funny. Well, that happens sometimes, no doubt about it. I sympathize. The question needs to be asked though, "Why were the stories so unfunny when you did have time?" I imagine that this is an unanswerable question. Animation writers believe there's never enough time.
On the other hand, maybe they're right. The kind of shows they come up with defy funny writing. How do you come up with a laugh-out-loud funny story for a typical writers' series premise like: "Six grade school friends (the skateboarder dude, the minority computer whiz, the perky intelligent girl, the over-eater, the inventer and the beautiful Bratz-type girl) solve problems in the school by making their fellow students aware of the importance of being themselves"? Is it even possible to write a funny story for a premise like that?
I do have advice for writers (hopefully artist/writers) who are lucky enough to work on short cartoon projects where humor is at least possible. If you need to write quickly then:
BUILD YOUR STORY AROUND A FUNNY SITUATION.
One character accidentally sits on another's hat and crushes it. He tries to apologize but the situation strikes him as so funny that he laughs instead. Believing the first guy is unrepentant the owner of the hat grabs a pair of scissors and cuts the other's tie in half, The two retaliate against each other in ever escalating steps till they're demolishing each other's houses. That's a funny situation, stolen from Laurel and Hardy. "King Size Canary" was about a funny situation. Funny situations are easier to write than plots, and they can easily be extended into stories. All you have to do is figure out how to get your characters into the situation and that's easy.
What some writers don't realize is that comedy isn't there just to get a laugh...
COMEDY IS AN ORGANIZING TOOL.
If you commit to writing a funny cartoon it's amazing how many story structure problems just vanish. In the example above the set-up is obvious: two friends meet in a kitchen and declare their undying affection for each other, an affection that would surely stand up to any test, no matter what. They then proceed to wreck each other over an accidentally crushed hat. If you have a good situation (sketch) idea the set-up practically writes itself.
In other words...
YOU CAN WRITE FASTER BY COMMITTING TO COMEDY.
A funny series premise is easier and faster to write for than an unfunny one. Complicated, writerly series premises are inefficient. They cost more to make and they're harder to write for.
BTW, I don't mean to imply that all stories should be written exactly the way I described here. There are lots of ways to go.