A few years back getting a lot of cavs in my army (200+ elite bounty hunters) always made the game lag wherever the cavs are. I've been playing an infantry based army for quite awhile now so I'm not sure if it's the issue. The game is perfectly playable when I do 200 battle size and gets progressively laggy as I turn that up. You could try lowering graphics/searching the internet for optimization but i'd say the battle size affects most of my problems and I just set everything else to on or middle ground option (if there's high low and medium i choose medium. I also set corpses to permanent and ragdolls to around 150 so you might want to look at that too.

Edit: The cavs issue persisted throughout the battle and would only dissipate once the number of units drop down to like 50 vs 50

CPU handles the game bad, probably the calculation of hundreds of AI cavalry