IMO, the US will almost certainly make up a lot of ground on the deaths per capita metric (or leapfrog entirely), given the explosion of new cases starting in June and the lagging growth in deaths that comes after cases. Scroll down to the "Deaths" section with the weekly per capita "heat spectrum" of sorts, and the US recently has reached red hot over the past few weeks, while European counterparts have been much cooler: https://www.nytimes.com/interactive/2020/world/coronavirus-m...
That helps, but at least for Spain and Italy, the hypothesis seems to be that the virus was already widely spread by the time lock-down started, so a lot of transmission still happened during lock-down until it got under control. I think it's the same for UK and France. but I'm less familiar.
Other EU countries like Germany, Austria or Poland fared much better than Spain, France, UK and Italy.
Maybe different cultures (e.g. how people relate, how compliant they are, how many people willingly ignore measures, etc.) also play a role.
This is totally at odds with the data. The UK, France, Spain, and Italy, among others, have higher deaths per capita than the US.