The old—even ancient—cities of Europe avoided the mistake of suburban building and have a naturally high-density shape from having been built before the automobile.

It's not visible in the center of cities, but an urban sprawl has developed on the periphery of cities. Such sprawls are similar to those existing in the US (car mandatory, few or no services, little public transportation).

the UK has the green belt system to try to prevent this happening

https://en.wikipedia.org/wiki/Green_belt_(United_Kingdom)

if anything it's too successful

As an aside (with apologies for sloppy definitions), I understand that the UK is about 0.1% densely built on (more than 80% of ground covered by artifical surface), and about 5.5% urban.

I understand this is much, much less than people typically think, which isn't just an interesting fact; people base their opinions on development on the unchecked feeling that the UK is 50% concrete.