Even the American cities that were designed after cars became common can be pretty intense in terms of traffic, though.
I took cello lessons when I was in school. My cello teacher lived in a neighborhood in Houston that (according to Google Maps) is 10 miles away from where I lived and should take about 15 minutes driving. However, since the route from my parents' house to my cello teacher's house went through some of the busiest freeways in the city (including one stretch that always had very heavy traffic even outside of rush hour), the trip generally took me an hour. Granted, I was going to cello lessons and returning home right smack in the middle of rush hour, but then again, rush hour took up about 6 hours of each weekday. I'd say the average trip time for that route outside of rush hour is about 30 minutes. That 15 minutes Google Maps quoted is what I'd expect it to take at 2:00 a.m., but that's about the only time of day it would be realistic.
I think the difference is that, unlike London and other European cities, there's no viable alternative to driving in a city like Houston. There is public transportation available, but it wouldn't be faster or easier than driving, so people don't use it if they don't have to. Using Google Maps again, taking the bus from my parents' house to that neighborhood would have taken over an hour, and I'd have had to walk about a mile on each end to get to and from bus stops. Even without a cello to transport, no way would I have done anything but drive if I had the choice.
So in short, I don't think it's so much the design of the city (because even cities that are designed with cars in mind can be bat-poo crazy to drive in) so much as the lack of viable mass transit in some places. In places where the public transportation is easier and faster than driving, people use it. There are just a lot of cities that don't really have that.