The problem in your case is that tiles 'remember' what cultural influence civs leave. You can see this number in one of the mouseovers. The exact formula is complex, but in short it means that if a certain tile has been accumulating e.g. Egyptian culture for thousands of years, its memory of the Egyptians will be rather high.
If you come, and capture the city, it will only grant you the city square itself. The other tiles will remain under the influence of the Egyptians, and if they still have cities nearby with culture reaching those surrounding tiles, those tiles will be within Egyptian borders. If you build up culture through a GW, you will increase the amount of culture that each time is added to the tiles next to you, but it can still take a while before you control them.
(it is more complex than just comparing the memories of different civs)
What to do? make sure no Egyptian cities can influence those tiles... if they can't fall within enemy borders, it will go to you as you do influence them.