London is a big city. The "official" population of London is 8.5 million, but if you add in the "almost connected" burbs it's probably really nearly 10 million. In this area, there is an average (which is falling year on year) of about 190 homicides per annum. For a similar population in LA County, the number averages about 650 and is rising.
For a similar population in the NY conurbation, the figure is about 350.
So, you are less likely to be murdered in London than in NY or LA.
If you compare murder per head of the population, you are 4 times more likely to die in Paris than in London (and that is excluding last years terrorist attack).
So you are safer, generally speaking, in London than in most big cities.
BUT there are places, and a number of them, where I would not walk alone at night in London (and I'm a bad-tempered big unit who's been around the block a bit).
It's a big city with all the big city problems, although perhaps fewer than you might believe from reading the newspapers.
A few more things that might surprise you.
You are **50 times more likely to be mugged in NY than you are to be mugged in London.
You are **100 times more likely to be shot in LA than in London. (They use knives here instead of guns, but fortunately, fewer of them.)
During the height of the "troubles" - the bombings and shootings in Northern Ireland, Belfast was still, statistically speaking, a much safer place to live than LA.
Still, Donald Trump says it's unsafe to visit London - I guess he stays away from all major cities.
** These are approximations from (about) 5 years ago - I haven't seen figures recently and there isn't enough data out there for me to recalculate them (and I'm too lazy anyway).