Here is my strategy to prevent culture flips after conquering a city.
First, end the resistance in that city. If it's a big city, you will need more troops. If the # of resistors are not decreasing, get your troops out of the city, it will flip back. They can always retake it later. Once resistance has ended, the real work begins. You don't need to leave very many units in a city after resistance has ended, usually 1 or 2. Leave some units outside the cities that are a greater flip risk. Also, try not to leave a healing army in a newly conquered city.
When I go to war I always make sure I have lots of gold for cash rushing, which usually means slower research. First, rush a temple (or library if scientific). Starve the citizens each and every turn. I turn them all into specialists. I used to just leave them as entertainers until I realized that this was stupid. Turning them into taxmen or scientists instead was much more useful (or engineers in C3C after replaceable parts). After a culture building has been built rush workers. Rushing 1 worker from scratch costs 80 gold. So the city will lose 2 pop points per turn. Once the city reaches size 1 there is very little chance of having it flip.
I don't know how well these strategies will work on Emperor+. I play at Regent or Monarch. I usually don't have much culture early on because I concentrate on military, but once the ball gets rolling those rushed temples help keep me ahead in culture. Also, my troops are almost always attacking and pushing forward, so AI counterattacks aren't too much of a problem.