Really the U.S. has a violence problem.
I don't really know the exact figures, but it was a chart considering the amount of murders per year per 100.000 people... The U.S. was at a murder rate Europe (on average) had back i nthe 1500's and the report's conclusion was that, going from those numbers, the U.S. has a pre-civil society. That's a bit of a shortcut in my opinion but it does illustrate.
There are a few minor exceptions but the deats in most of Western Europe (comparable in culture to the U.S.) there, for the by far largest part, are caused by two reasons: People who severe mental disabilities, where you see usc tragic cases that father murders his entire family) and criminal liquidations, about which the police in general doesn't really give a fuck about.
This is not at all the case in the U.S.
Considering population density in this regard doesn't exactly prove your point as Europe is more densely populated than the U.S.
Even if you would only consider city density the U.S. would lose. As a small example, take a pretty generic city in the U.S., for example Cincinnatti, Ohio.
A city with roughly 298.000 people.
Now take a pretty generic German city like Bonn, Westphalia, with roughly 318.000 people - slightly larger, but comparable.
The area of Cincinnatti is about 206.1 sq/Km
The area of Bonn is about 140
There are some really obious explanations to this, though.