On another note.
About the armies of Byzantium beating the turks this could (sadly or not) never happen. Stave the off, yes. The problem is that light archers on horseback was the A-bomb of the middle ages. No weapon could beat it if all else was equal. You simply have to stack the deck very much in your favor to beat them.
War is about mobility, attacks and defence. Lightly armored and on horseback gives you superior mobility (Compared to walking, or riding while armored). Bows give you superior attacks (compared to all kinds of melee weapons, that cannot reach you)
Defence is lacking, but easily compensated for with speed in evasion and range of attacks.
In short you have to rely on bad leadership to score a decisive victory against horse-archer nomads. Else you cannot kill them, as they run away, and you cannot take their homeland, as it runs away
If you have good leadership you can make feints and use tricks that might fool the nomads into bad moves, but history would say that a gullible leader is a bad one, bringing us back to point one.
The only upside to this is that nomads basically need good leadership to win any decisive battles on their own, as they are quite inclined to loot, pillage and feud about loot or old scores if without strong leadership, and discipline is necessary for sieges.
Still, the WRE fell because of Attilas hordes, and the ERE were briefly saved by Timur Lenks hordes. Hordes simply put are the movers and shakers, but not hoarders
Nomad empires are the exception, not the rule, and thus historians have looked down their collective noses upon them, and therefore their military superiority is still rarely acnowledged, which I guess is my point with this rambling, that the Byzantines would loose the anatolian lands sooner or later, as the land was needed by 'immortal' armies and their herds.
