The website will be liable for the copyright infringement as if they would have uploaded the material themselves if they do not implement a good filter whatever that is.

It seems as though they would have a hard time doing this. If the website is based in America, they can't take it down. They probably can't extradite either; the rule is typically that it has to be illegal in both nations.

How can they enforce this?

If the website is not hosted in the EU then the law will have no effect.

I think many websites will simply move to other parts of the world. That is what I would do anyway.

That's what I figured. The worst part about it is that it won't affect shady cyberlockers whose business models are rather suspect and often not known to tax authorities, but legitimate sites that do business in the EU will be impacted.