In the US, there is no exception to copyright based on the length of the clip used. In practice, though, copyright holders often ignore infringing uses if they are short and not commercial in nature -- but they don't have to, and they can change their minds at any point in the future.

If you're talking about using clips in an advertisement, though, copyright holders are much less likely to look the other way.

> Is there any way that smaller websites can play short snippets (10-20 seconds) of copyrighted famous songs on their website?

Yes. The legal way to do this is to buy a license. Usually, BMI or ASCAP are the entities that deal with this. You should contact them for terms. You might also want to consult a copyright attorney for authoritative advice rather than listen to an internet random like myself.