Is that just because Americans spend a lot more time in their cars than Europeans do?

No, per km Europeans are safer. Road design is safer, cars are safer.

I'm not sure it is so simple. Some states have death rates in line with western/northern Europe (eg massachusetts has 4 deaths per billion km, similar to Germany), but road design and cars driven in massachusetts are highly similar to states with a larger death rate(south carolina, looking at you).