I had to deal with this soooo bad on my recent emperor Mali game. Eleanor's France Dow'd my ally Norway, and so my other ally Inca and I went to war with her (it was quite the love-triangle we had going on). Now she was way ahead of us all in culture and had been culture-creeping across the whole continent with those great works. She'd eaten up most of Brazil and parts of Arabia already, and if we didn't stop her, I might not have allies left by the time my starship arrived at its destination. Well, culture might have been her advantage, but economy, tech and military might were mine, not to mention a juicy new fully upgraded GDR that needed some field testing...
View attachment 541886
But as it turned out, her culture pressure was INSANE the moment I started capturing stuff. Even putting a governor in my newly captured cities wasn't enough. I realized that I'd have to take every single one of her original cities within a handful of turns or I'd start losing them back. So that's what I did. Between my airforce and naval bombardments, along with my GDR and modern armor I blitzed through France like Hitler in 1939. I was taking a city every turn, stacking policy cards for maximum loyalty bonuses, parking a governor in each one, and generally just bulldozing everything. Honestly I think that's the only way to take down a cultural monstrosity like Eleanor's France. Any slower and those cities would have been revolting faster than I could take them.
One other tip to consider: If you can't blitz that fast, try this - bombard your enemy's city defenses down to zero, then bypass the city and go on to the next. Station a melee unit near each city that you've pacified like this, and then, once you've got several defenseless like that, take them all in a single turn! Have your melee units move in all at once, taking every city that you'd previously softened up, and then boom, no more loyalty issues cause you got them all instantly.