Well by virtue of the fact that you reside in Europe, you're subject to European laws. And by using a server in the US, you're subject to some US laws as well.
But if you're in the US, and your servers are in the US, you shouldn't have to obey European laws. European site visitors are no different than physical European tourists in the US; the owners of those US museums and tourist sites don't need to obey European laws just because they have European visitors.
But if you're in the US, and your servers are in the US, you shouldn't have to obey European laws. European site visitors are no different than physical European tourists in the US; the owners of those US museums and tourist sites don't need to obey European laws just because they have European visitors.