I realise that this is one of those questions which might be answered differently depending on the state. I also realise that there are special cases such as the boroughs of NYC. However, I suppose/hope there are some common features of American local government that might allow for a more general answer.
As I understand it, local government functions in the US are, by default, carried out by the counties, but when a particular area within a county is incorporated as a city, then that city will carry out those functions autonomously. The territory of the city remains, on paper, part of the territory of the county, but the county retains no functions for the city, so the inclusion of the city in the county is only nominal. Is this correct? It’s the only way I can make rhyme and reason of the fact that it is, apparently, not rare in the US for a city to straddle the boundary between two counties. Does this mean that, outside incorporated cities, there is no local government other than the county, and consequently no elected local decision-making bodies? It also means that there is no sort of supervisory power of a county over the cities in it?
(The reason I’m asking is that in the jurisdiction I grew up in, Germany, counties and cities have different powers. A city might carry out some functions, and the county it belongs to carries out other functions for the cities in it - unless the city is independent, meaning it is part of no county whatsoever and combines city and county functions at the same level.)