tons of elected sheriffs in the US, usually at the county rather than the city level. for example in the rest of New York State outside of the city, there are elected sheriffs. but there may also be police departments associated with towns or cities. like so much of US government it's a complicated federalist mess.

i don't know how it works on average but there are a few high-profile cases of elected sheriffs seriously abusing their office. Joe Arpaio is the most famous. he had a strong base of support from people in the county, and wasn't afraid to use the powers of his office to target political opponents, which let him stay in power for many years.