Yeah, TC, gotta have those stakes in open field, or an all cav force, to have much chance. But the bridges are great, just make sure you get them coming from one side only. I've pulled off battles with forces coming from both sides of a river, but only when the reinforcements are infantry! They are slow enough that you can sometimes crush the original force, then shift sides of the river. Or successfully screen off the reinforcements with something mobile until you can shift. But not against horde horse archers and cav in any number. Too fast and they can shoot on the move.

Walls and bridges are the best bets. Once you know the battle terrain well, there are some passes that will work too. I am still at the guessing stage there, but I've seen some places where they can easily be beaten in the mountains too. Heh, mostly because I tried to attack them there and decided that was a BAD idea. Lots of cliffs. Places where approach is worse than with bridges. But you really need to learn those spots by playing a lot of battles in the mountains there.

Jerusalem has some great cliffs and ridges around it too, but you need to learn those also. And be sure you're on the right side of them.

I've only fought them as Turks to date (working on getting that far with Russians now). My JA had the stakes too. I managed one nice bridge battle with an infantry force, but couldn't tempt them into another. All the rest were HA armies, mostly hitting and running, taking the "loss" to whittle them down until I could crush the remnants. But I did manage an ambush once, and that was a blast. Single full stack all lined up nicely on the road, with high ground on one side, cav in the van, infantry (mostly archers in the rear). I set up HA on both high and low sides of the foot troops, but kept the rest and my general on the high side of the cav. The cav got shot up from range and upkill, the infantry got charged to force them to skirmish as my HA shot them to pieces. That was one full Mongol stack that took a licking by my Turkomen and Sipahi. Other than that the only place I beat full stacks was on bridge battles. The rest was attrition. Their HA are just better than Turk HA. Mamluks might whup them though.

I never got the Mongols to attack a single city though. All had stone walls, at least, and good garrisons, and they'd approach and then leave. Never even tried to besiege. Next time I will use smaller garrisons. It's quicker to lose a city or two and retake them than spend decades chasing the Mongols through the mountains trying to force battles in favorable terrain. Get them to attack the walls and attrit them that way.