I'd hardly say that the European forces were torn up. They certainly suffered heavy defeats, but they also inflicted some heavy casualties in a few of the battles. At Mohi, for instance, the Mongols sustained significant casualties, and at Tursko the Poles won the initial battle before spreading out to loot the Mongol camp.

In general, the European forces were either of a size with the Mongol force or else smaller. It may even be that at the Battle of Mohi that the Mongols had more than half again as many men as the Hungarians.. At the other battles of the period, there seems to be a similar force parity or disadvantage on the European side.

This is important because most of the European forces were not cavalry, but infantry. No precise ratio can be determined, but cavalry typically formed between 1/3 and 1/6 of an army in this period. The Mongols, on the other hand, were entirely a cavalry force. This gave them both superior tactical mobility but also meant that if the knights were separated from their supporting infantry they would be considerably outnumbered.

To this we need to add the various political differences and rivalries between the various constituent parts of the European forces. There were often long standing feuds and enmities that, while theoretically laid to rest in order to fight the external threat, nonetheless simmered in the background and caused issues.

Using Mohi as an example again, the Hungarian king had been confiscating land from his nobility, had forced them to stand when in council with him and his alliance with the refugee Cumans had caused problems for the nobility, with the king supporting the Cumans in disputes. This culminated in a riot that saw the death of a Cuman khan and the Cumans splitting from the Hungarians and pillaging the countryside as they left, which reduced the nobility's ability and willingness to send troops to aid the king.

During the lead up to the battle Duke Frederick II of Austria and Styria - who had arrived to help fight the Mongols - and archbishop Ugrin Csák both disobeyed the directive of the king to avoid battle, with mixed results. The king, meanwhile, despite his early caution, did nothing to rouse his men and get them into fighting order on the morning of the battle, which may have cost them the battle.

As you can see, political disunity played a big role in both the European countries ability to raise troops, but also to command them. Without effective leadership, it becomes almost impossible to prevent the impulsive charging of a seemingly retreating enemy by the knights and other cavalry. Once away from their infantry, the knights could be engaged on three sides or split up even further, and without cavalry to protect them the infantry were easy prey for the Mongol cavalry.

In short, the Mongols had the numbers, the tactics and were not made up of a patchwork of disunited forces.