I’ve been to the US at least 10 or 11 times now (mostly for work).
Of the probably 60 or so people I’ve been with on various trips, I’ve only known one gun incident – they went jogging in Atlanta, and obviously went into a neighbourhood that was a bit less tourist friendly than some, and were told by a man brandishing a gun that they should turn round now and go back to where they’d come from, followed by shooting a bullet over their heads as they returned the way they came, at more of a run than a jog to say the least.
Really, it is very segregated by area (even by state as the graphs above show), so other than random nutters, as long as you stay out of the dodgy areas you’ll probably be fine. The only thing that I am aware of in US cities is that the distance between pretty safe tourist area and proper scary dodgy area, can be extremely small, like a couple of blocks walk, and when an area is dodgy, it can be way more dodgy than most UK dodgy areas. I’ve been caught out a couple of times having an explore and had to hurry back to a well lit street!
Having said that, Europe can be scary too – scariest place I’ve been for that has to be Athens in Greece – literally 100 metres from lovely tourist cafés to a street full of smackheads literally cooking it up on spoons and foil on the street corner.