University of Leiden study. Buried toward the bottom
of this webpage. Also
alluded to here.
EDIT:
The reason that the U.S. is not in the top 10 for "highest levels of violence in the western world" is likely the same reason as for any "higher gun violence". That is... We use firearms for self-defense about once every 13 seconds. Of these instances, 15.7% of the people using firearms defensively stated that they "almost certainly" saved their lives by doing so. Successful self defense shootings are included in the "homicide rate", as are police shootings, criminals shooting criminals, and suicides.
I wouldn't count anything in the last sentence as a violent crime. Successful self-defense with a gun is obviously not a crime (maybe it is in Canada, from what I've read here), although it is certainly a "violent gun use". Police shooting a criminal is also not a crime, hopefully. Criminal shooting a criminal (mostly gang/drug related) strictly speaking is a crime, although I would call it more of a "public service". Criminals shooting criminals is by far the most frequent use of "gun violence". It heavily weights the gun use/violence stats (anywhere from about half to 90%). Suicides a crime? If so, who will you prosecute?
The statistic that means the most to me: Firearms are used 60 times more often to protect lives than to take lives. <--In the great majority of cases, the gun successfully defends without firing it.
(Source/citation for most stats: last link listed above.)