(In fact, in America, the last few decades' worth of safety improvements to cars have been almost entirely offset by an increase in miles driven; despite their much safer cars, Americans are only slightly less likely to die driving today than they were in 1990 [1].)
The bigger mystery is why Americans are now far likelier to die in their cars than Europeans even after you control for the time they spend in them. That didn't use to be the case. Until the mid-1990s, Sweden (like almost every other country [2]) had a higher rate of traffic fatalities per mile driven than did the United States. Those rates have declined markedly all around the world, but much less in the United States than elsewhere.
My guess is that the divergence comes down to diverging patterns of living. Over the last few decades, American cities have sprawled all over the landscape, so Americans not only drive more, but their road infrastructure is built to facilitate daily travel at high speeds over long distances. A great many Americans drive 30+ highway miles each day between their suburban homes and city offices, but that sort of commute is rare elsewhere.
0. http://en.wikipedia.org/wiki/List_of_countries_by_traffic-re...
1. http://www.planetizen.com/node/68200
2. See, e.g., statistics showing that fatalities per vehicle mile were lower in the United State than in many European countries in 1991, https://www.fhwa.dot.gov/ohim/hs93/Sec7.pdf [PDF; see page 4]