So, you designed a Flash game all by yourself? Want people other than your family and friends to play it? What about people all over the world? Probably you have seen Miniclip. In case of the unlikely event that you don't know, it's the best online gaming site on the web. All are free, no downloads, and a perfect site structure. And I'm not paid to write this. And I'm not an employee.
So, I haven't done this, but I noticed the small link in the bottom right corner of the site that says Submit Game. Now that's how companies get their games there. Also, this shows that you can send your games to them.
When you click it, it turns out to be a mailto: link. If you don't use an email program, just send an email to "submitgame@miniclip.com". Well, I think you should just send in a request to them and they will follow up.
You can probably find other websites that can accept user-designed games. Run a Google search. You'll find out.
Good luck. And remember, they will probably have their rules and criteria for publishing the game.