Comicspace, the "social networking" site for comics fans and pros, has just added a feature by which members can post images of their own comics and art. There is some very cool stuff on there. Here's the link:

http://www.comicspace.com/comlist.php

I recommend just browsing and clicking on anything that looks interesting. There's superheroes, but there's a whole lot more, too. Do non-nerdy people realize how much breadth of variety there is in comics? There's plenty.

I'll point out one story I just read tonight, "Untitled", by Winston Rowntree. The whole short story is here. It starts off as a charming slice-of-life vignette, then has a big turn. The twist is jarring, but it makes sense in the end. It's twelve pages.
http://www.comicspace.com/winstonrowntree/comics.php?action=gallery&comic_id=963