Because controlling the influx of weapons, legal and illegal, in European countries is much easier than it is in the US. The smaller a country is, the easier it is to control the flow of firearms; you have less area to patrol, your borders are smaller, and there's a greater overall level of population density
Logistically speaking, controlling guns in the US like they do in Britain would require ludicrous amounts of manpower and massive government expenditure, its practically impossible. We have a massive amount of land, plenty of backwood, out-of-the-way places where gun runners can hide themselves and their caches of illicit weaponry, and our borders cover thousands of miles, at the very least. There are numerous points of entry, and lots of places to hide, so trying to prevent anyone from getting guns in the US is simply impractical.
So, instead of trying to do something that'd likely mean massive tax hikes (which would piss people off REALLY bad) to cover a very impractical set of laws, they, instead institute a policy of background checks and waiting periods. Does that eliminate gun violence? No. Does it mean criminals won't get guns? No. Criminals don't obey the laws for rape, murder, and theft...why would they obey gun control laws? PROTIP: They don't. They never do. Anyone who wants a gun can get one...so, the question is, do we spend massive amounts of time and money in a futile attempt at an impractical measure to control illegal arms, or do we institute policies that help weed out dangerous people from legally purchasing arms, while allowing decent citizens to procure them for sport or self defense?
Take a look at the war on drugs. Its almost the exact same thing: they're trying, desperately, to control the flow of drugs into the country, spending massive amounts of time, money, and effort in an attempt to weed out any and all illicit medicine. Are they succeeding? ...So very little, its practically unnoticeable. Instead, we have a significant portion of our prisons dedicated to non-violent criminals, we make criminals out of otherwise decent people, and we force the money that goes into purchasing these things, which is almost laughably easy if you ask the right people, into the hands of much more dangerous criminals and drug cartels. Its a logistical nightmare, but the government continues spending millions of dollars in this futile endeavor because of the pressure put on by holier-than-thou assholes.
We've already got one pointless "war." I doubt we could afford another, even if such a stringent crackdown wouldn't end up a legal nightmare.