They do it with the Eurotunnel, though I can't remember if I saw anything but trucks and busses on it
The auto trains that go across Europe are much narrower and take a lot longer to load and unload. They are pretty expensive and only used for very long distances. Most of the time it makes more sense to rent a car at your destination.
Normal bridgdes where? They fit perfectly fine on the French loading gauge (max train dimensions), but the British one is much smaller on most lines.