Weak cavalry?

[Furious Mental]

"If you think infantry was so great with these shields why Gengis Khan used only horses?"

First of all, Genghis Khan didn't use "only horses". Where did you get that? Do you think he took all those Chinese citadels with cavalry? Of course not, he had to use Chinese siege engineers and Chinese infantry. Second of all, in so far as Mongol field armies were often comprised entirely of cavalry, the example is quite irrelevant to the discussion because Mongols were so different to Europeans in terms of culture and tactics.

Mongols, being steppe people, naturally tended towards mounted warfare. How could a warlord control the vast plains without horses? He couldn't. In the European context the relative slowness of foot soldiers was less problematic- take the example of a very warlike period in England, 1135-1154: any army was likely to be marching a very short distance to besiege one of the several hundred castles then in existence.

Notably Mongols also did not rely solely on the frontal charge; the charge generally came last after an enemy had been worn down with arrows and often coaxed into pointless and tiring attempts at pursuit, which often placed them in a vulnerable position.

"If you think infantry was so great with these shields why Gengis Khan used only horses? (the most of his opponents used them and even than they lost) Why the cavalry was the main force in Europe and America during medieval times and even after."

Your view of medieval warfare is simply wrong, there are no two ways about it. Infantry were the main component of the average medieval European army. Most medieval generals followed the strategy of the Roman theorist Vegetius, which was to avoid the risks entailed in pitched battles and instead try to force the enemy to abandon a campaign by gaining control of food supplies. In doing this one of the most important things to do was gain control of castles, so sieges were far more numerous than pitched battles. Cavalry are relatively useless for sieges. Where cavalry did fit into this sort of logistical warfare was chiefly using their mobility for foraging and harrying. They were certainly important in pitched battles when such battles occurred but were generally neither the predominant portion of the army nor could they just trample down any opposition. As is the case with any tactical unit, they were best in conjunction with others.

"Someone said that shield wall was good against horses."

Go and read the account of the Battle of Hastings in the Gesta Willelmi, which was written by William of Poitiers, who was William the Conqueror's chaplain and therefore can be taken to have gotten his information straight from those who were at the battle. He was unambiguous in stating that stating that a couple of thousand of the best cavalry in France could make little impression on the English shield wall, and only won by hours of skirmishing (in conjunction with archers and crossbowmen on foot) and tactical ruses. In other words- they fought like Mongols (and some historians theorise that in fact the knowledge of this style of warfare was derived from Frankish contact with the Magyars). And after the Battle of Hastings there was no small number of battles where the English fyrd, then at the service of the Anglo-Norman kings, withstood charges by continental cavalry again and again. Notably, on a number of occasions, the Norman knights in fact dismounted to strengthen the infantry formation; in other words the king guessed (correctly the results of the battles show) that his cavalry were worth more on foot than they were mounted.