British Bobby, welcome to The Firing Line.
Despite any apparent correlation in gun laws vs. crime, there is no real causation. This has been borne out in several major studies over the last decade. This includes studies that started under the premise that more gun control = less crime (See 2003 CDC Study
Fact is, there is no actual correlation between gun control and crime, whether we are discussing the U.S. or Europe. See Google Scholar
for more articles on this topic than you would otherwise think.
Given that correlation is a myth, I guess that the question I have, is what do you mean when you say gun restriction works in Europe and not the U.S.?