American cars were big way before they became obese. Americans have been more suburban and roads were developed before a lot of towns were, so roads are wide and big.
I love small cars. I’m happy to live somewhere with small roads. And I care deeply about living in a walkable city, but obesity is not the reason the US has big cars and the UK has small cars. Similar obesity rate. Different roads. Different fuel costs.
Obesity rates in the UK are not far behind the US. 32% vs 38% according to WHO. Unfortunately most western developed counties are well on the path to where America is regarding obesity. The key difference in terms of car sizes is size of roads, civil planning, gas prices, and marketing. Not obesity.