Actually, I thought it was the Turks that stopped the Mongols dead in their tracks. They were every bit the Mongols' equals in warfare.........and damned well knew it. Their first engagement (a nasty affair) ended in a draw, but it managed to halt the Mongols' advance. They never quite regained their momentum.
The Mongol superiority was in C3I. They could maneuver and control over wide distances with a skill that no other medieval army could match. The first defeat the Mongols suffered was not at the hands of the Turks. It was the Egyptian Mamelukes at Ain Jalut. Egypt specifically set about to create a body of elite heavy cavalry who could both outshoot Mongol horse archers and stand up to a Crusader charge.