Muslim heavy cavalry is already much improved from MTW, where they didn't even have barded horses and got shot up rather easily. Once you get a decent general and some experience for them, they will easily beat Catholic knights.

600,000 crusaders is definitely inaccurate. All military accounts exaggerate troop strengths and casualties, even today. Ever notice how in England vs France military history, France consistently loses 30% more men even when they win the battle? Medieval accounts are defintely unreliable unless proven otherwise. History is written by the victors, and accounts from different sides will disagree greatly. The Turks claim to have defeated 600,000 Crusaders, but the French claim to have slaughtered 200,000 'Moors' in Spain, the English to have beaten 200,000 French knights at Agincourt, the Greeks (Macedonians really) to have annihilated 2 million Persians at Issus, and so on. It's ridiculous, to say the least.

600,000 fighting men in Medieval Europe? That is more men than the entire Imperial Roman army under Augustus, auxuiliaries included. Logistically, they could have won simply by eating up the whole Middle East. A more reasonable estimate would be 100,000, including a large percentage of camp followers and other non-combatants. Opposing them were 20,000-30,000 Turkish fighting men at most. Feudal armies could be very large, but only a tiny fraction of them would actually be hardened soldiers (knights, men-at-arms, mercenaries like longbowmen). The rest just scream and run around a lot when the fighting men are defeated.