When I have time to go to a movie, I usually have to sit through several short movie clips (usually less than 10 min) before the movie starts. The makers of the full length versions of these movies have to pay to get their clips in the movie.
In my opinion, YouTube does not get shutdown because many of their clips are free advertising. If I go to YouTube and watch a funny clip of the Simpson’s movie, I am not more inclined to pay to go see the full thing.
Same holds for TV Shows. If I see a clip I like, I now want to see the whole thing, something that YouTube cannot conveniently offer. This leads me to the normal commercial channels to purchase the content.
If Napster had only offered short segments of songs at sub digital quality (similar to what YouTube does), I think they would still be around today (though no one would really be interested in a site like that).
In short, they don’t get shutdown because it is free advertising. In the case were the advertising is negative they send a copyright notice and get it removed.