Ordinary archers outrange horse archers,even an ordinary archer unit can beat turcoman or ordinary HA's.But mamelukes,Faris,Boyars,Byzantine Cavalry and Steppe heavy are a problem,you'll need to combine fast cavalry with foot archers.Crossbows and mounted crossbowmen are good counters to HAs too.Terbizond archers,Nizari,Turcoman(and other turkish hybrid archers),Bulgarian Brigands,Pavise arbalesters and Pavise Crossbows handle mounted archers pretty well.Alan Mercenaries become plentiful when you reach the steppes,you can also recruit hobilars,they only need a small amount of infastructure investment and you can get them(also if you hae a steppe territory recruit steppe cavalry).