Medieval Armies that employed horse archers seemed to have a lot of success in battle, The early Hungarians raided most of western europe, the turks defeated the byzantines at manzikert, and Saladine's crushing victory over the crusaders at Hattin. These battles are usually used to illustrate how the agile horse archers were able to say out of range of their slow, plodding oppenents and pepper them with arrows until they either routed or were run down. However, I read that the slower, more heavily armed Germans were able to defeat the Hungarians at lechfield and the crusaders and Byzantines were able to do the same and score many victories against Turkish and arab horse archers. How were the heavily armored slower armies able to to pin down and defeat armies using horse archers?
The Byzantines used Horse Archers as well as Cataphracts. They adapted their armies to be very Calvary based due to their fighting with the Sassanids.
That said horses were very expensive to buy and maintain and horse archers were not all that effective in the heavily wooded German regions. Horse archers need a lot of room to maneuver to be effective, so they were usually beaten when their amazing mobility could be restricted.
Tactics to reduce the mobility of horses and horse archers at least in the early medieval period would have been trenches dug that would be covered by the dust created by an army.
In the Battle of Jaffa, during the 3rd Crusade, crossbowmen proved very effective against Saladin's horse archers.
Saladin had taken the town of Jaffa from the Crusaders, but the citadel held out. Richard the Lionheart attacked the town from the sea and drove the Saracens out.
Saladin then attempted a battle with his horse archers to retake the town. The clash between the two armies took place in the fields outside the walls of the town.
Richard's army consisted of about 54 knights, a few hundred infantry armed with spears, and 2,000 Genoese crossbowmen.
The crossbows shot the horse archers to pieces. Saladin was said to have lost 700 men and 1500 horses. Crusader casualties were two men.
It was this victory which led to the peace which concluded the Third Crusade.
A fairly good example of a battle won by medieval European armies against horses archers would be the battle of Dorylaeum (1097) fought by the armies of the first crusade against the Turks led by Kilij Arslan.
The battle was described in the Gesta Francorum et aliorum Hierosolimitanorum ("The deeds of the Franks and the other pilgrims to Jerusalem"), whose author was a first-hand witness serving in the army of Bohemond of Taranto and is available here: http://legacy.fordham.edu/halsall/source/gesta-cde.asp#dory
The vanguard of the crusader army was attacked by an important turkish force of horse archers. Taken by surprise, the crusaders were unable to engage the opponent skirmishers (although their heavy armors protected the knights from most arrows, their horses and less armoured foot soldiers were not as lucky), and Bohemond of Taranto had to organize a somewhat desperate defense.
As the day went on, the Turks became more agressive, trying to press their advantage in order to break their opponents morale. However, in doing so, they became vulnerable to a surprise attack: the rear guard of the first Crusade, led by Bishop Adhémar of Le Puy , made it into the battlefield at this moment and managed to close the gap with the bulk of the turkish force.
In melee combat, the average french or norman knight was extremely favoured against the lightly armoured horse archers. By outflanking the turkish troops, the crusaders won the battle and opened the road to Antioch. Proper use of terrain and the element of surprise were the key factors to most victories the first crusaders won against forces using large amounts of horse archers.
Sources:
Z.Oldenbourg, Les Croisades, Gallimard, 1965, p 147-151
Louis Bréhier, Gesta Francorum et aliorum Hierosolimitanorum, 1964. French translation.
R. C. Smail, Crusading Warfare, 1097-1193, Cambridge University Press, 1995