My I - as usual you are comparing apples and bricks.
Definitions of crime vary by country and legislation. Measuring 'crimes' is meaningless without a very long list of caveats and exceptions.
Murder/Homicide is the only statistic that most countries collect on a similar basis but even that can vary by jurisdiction.
Per 100,000 of population:
Murders? The US leads the UK by a long way. 5.35 per 100,000 to 1.20
https://en.wikipedia.org/wiki/List_of_countries_by_intentional_homicide_rate
Deaths by firearm? Not surprisingly the US leads the UK by a long way. Firearms are not so easily available in the UK so the difference is obvious. That doesn't mean the UK is immune. It means that obtaining and using a firearm is more difficult in the UK.
If you count murders alone the UK is much safer than the US. Anything else? It depends how each country compiles its statistics.
I find it interesting that Switzerland is a free gun society yet they don't have these problems. Gun ownership there is on the decline, but that is purely voluntary. The Swiss have simply decided that they don't want them.