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If there's the right districts, run Bread and Circuses. Set policy cards that boost loyalty.
And ultimately, make absolutely certain that you're not leaving a huge gap between your territory and the territory you're conquering, or you will have to take multiple cities at once to have enough loyalty pressure to stave off rebellions.
Among all of those factors there are two that are especially important, governors and population pressure.
Governors add 8 points in your favor. That's big.
Population pressure is the basis for calculating loyalty. Every city within 9 tiles of a city exerts loyalty pressure on that city. The pressure is greater the greater the population of these other cities, and is also greater the closer they are to the target city. The target city's pop counts the most, because it is the closest you can get to the target city. The loyalty pressure is modified by what sort of age you are in compared to the other civ. Pop points in a golden age exert twice the pressure, dark half the pressure.
The practical difficulty for conquest is that you often find yourself taking a low pop frontier city from your enemy, then capturing it reduces its pop further. This low pop city then Has to withstand the loyalty pressure from your enemy's larger core cities. This is a problem.
The way around this problem is blitzkrieg. You have to take enough core cites from your enemy soon enough after the frontier city or cities that you capture first that they don't have time to revolt before you take the core cities to create a viable nest of loyalty pressure in your favor. Taking one of their high pop cities is a twofer, you reduce their pressure and increase your own.
If you are planning conquest into a dense nest of enemy cities, you might get by with arranging all the loyalty boosts mentioned above. If that won't do it, you just have to wait and plan for a blitzkrieg.
Also attacking in fast succession is an option as well, but that needs to to outtech the opponent with you siege units compared to their walls. With that strat you still might need a gouvernor to hold your first city to extend the possible loyalty flip up to like 4-5 turns, but usually that's more a lategame strat, where you often attack on multiple fronts in my experience.
Overall with the first strategy being an option, loyalty isn't really a problem in most situations, so you can keep more useful cards in your policy slots, especially those who give a better yield long term (like +25% experience gain for attackers or obviously many economic cards), as well as cards, which allow you to attack faster (like +1 movement or x attack strength within friendly territory), thus solving the loyalty problem faster.
Though i would always recommend to have at least one gouvernor available, preferably Liang or Magnus, if you don't need their buffs right now.
The best method is to take cities in triangles so 2 cities are always benefiting the other in every direction. Cutting swamps and rain forests for population can help you rapidly stabilise a city. Lastly you could just let the city either flip or burn it to the ground.