Yeah, over here you could do that. In every hotel room, or private bathroom. Kids learn not to use a hair dryer while in the tub and stuff like that. Surprisingly very little ever happens.
In more modern houses than ours there is a special fuse (ground fault circuit interrupter?) that'll click out if any power runs into earth. It gives you some safety.

I don't remember - don't they have sockets in the bathrooms in the US? Where do you plug in your razor? Or the electric toothbrush?