Did not seem that Western European armies stood up to Turco-Mongol military tactics such as the horse archer. Did these European armies ever learn to counter those tactics? The Slavs (Polish) had a few successful fights against the Tartars, maybe I'm wrong.... (referring to either Mongol invasion of Rus' or List of Mongol-Tartar raids?).
Western Europe did have horse-archers; it is a false perception that they did not have an answer to the mounted archers of the Horselords (Although whether they were as effective as their steppe counterparts is a different question).
The Roman Empire fielded mounted archers who were called Equites Sagittarii. After the fall of the Western Empire, the Eastern Empire continued the Roman traditions of mounted archery in the form of the Archers of the Kavallarioi.
As for why didn't the post-Roman era Europeans learn the tactics when they came in contact with Altaic Horse-Archers, they most certainly did in a fashion. German and Scandinavian Medieval Armies employed mounted Cross-bowmen. The tactics adopted by the mounted-crossbowmen to counter the lightly armoured Altaic mounted-archers were to gallop headlong, fire a devastating volley at close range and then charge at their enemies with swords or lances. Innovations in Crossbow technology such as goat's foot lever and ratchet Cranquin allowed the Crossbowmen to reload fast enough to fire more shots before engaging the enemy in direct combat.
Another tactic was fielding more heavily armoured cavalry which could withstand the arrows of lighter Horse Archers. Russian Druzhina cavalry was developed as an answer to the Tatar horse archers. Altaic people themselves adopted this tactic later against European Horse-Archers e.g. The Timariots and Kapikulus of the Ottoman Sultans were as heavily armoured as contemporary Western Knights to withstand the volleys of Hungarian, Albanian Horse-Archers.
But of course, using crossbows and mounted archery required skills and experience which the peasants of Europe (Bulk of Medieval European armies) did not have. That's why Mercenaries were preferred in that role. It was soon noted by the Europeans that Horse-Archers were ineffective against massed foot cross-bowmen who could fire more rapidly, accurately and at a longer range than the mobile but larger and slower-firing targets that the horse-archers presented. That made Horse-Archery very outdated and eventually obsolete with the invention of firearms (Which in turn prompted the Steppe Warriors to exchange their bows for firearms).
So in summary, Western European response to the mounted Archers of the steppe was:
- Heavy Cavalry
- Mounted Crossbowmen
- Mass deployment of foot Crossbowmen