but didn't go to the Bronx or Harlem. Others have told me that these places are not as safe, but we had no reason to go anyway.
It's a big city, like London. the same rules apply.
I have not been to the Bronx but I have been to Harlem a few times and to me it feels reasonably safe; we were two middle-aged white guys wandering around looking at architecture, looking at the neighbourhood and hitting a few bars; we were sensible and we felt comfortable. Your point about London is a good one.