Do you think catching nomads and fighting them are separate matters? Battles are fought on the strategic level, not just who would break through who after formations are already deployed. The Safavid was able to stalemate the Ottomans and even capture their territory during Shah Abbas' time despite having vastly inferior sized forces because of superior mobility and logistics.
The Safavid weren't just superior to the Ottomans in a game of tag either; it was superior in field combat as well, which can be seen when in 1603, 6,000 Safavid cavalry routed 5,000 Ottomans even though the later also deployed a wagon laager and manned it with musketeers and artillery. The Safavids generally avoided superior Ottoman numbers and whenever confronted, they often won with cavalry charges throughout Abbas' reign.
In the battle of Chaldiran where the Ottomans supposed "demonstrated" their superior artillery and won the battle, the Safavid only had 20,000 against 70,000 forces. Even with vastly inferior numbers, the cavalry charge almost won the day, and both sides suffered heavy losses, and the Safavid only "lost" by withdrawing. This is also considering the fact that Ismail listened to his right wing Qizilbash commander to wait after the Ottomans deployed their cannons. According to Chase, had Ismail listened to his left wing Qizilbash commander who urged him to attack before the cannons were deployed, he might have won the battle (just like how the Jurchens were able to rout Ming armies before the later deployed their cannons at Sarhu).
You are giving examples of Hussars fighting European and Ottoman armies, and claiming they are better than nomadic cavalries and are the best in the world because of that. The Ottoman cavalry is not a nomadic cavalry and its far inferior to real nomadic Central Eurasian cavalries further east. The Safavid-Ottoman wars already demonstrated that.
You are implying that Lithuanian Hussars are better than nomadic cavalries. I have found no evidence whatsoever of this in the encounters.
According to de Beauplan, who served in the Polish army in the 17th century said this about the Cossack army:
"On horseback they are not the best. I remember having seen only two hundred Polish horse rout 2,000 of their best men. It is true that one hundred of these cossacks, protected by their tabor, have no fear of a thousand Poles, nor even of a like number of Tatars. "
Description, 13
This implies that the Cossack infantry with a tabor (wagon circle) have no fear of Hussars. The next comment "even of a like number of Tartars", implies that de Beauplan viewed the Tartar cavalry as even better than the Polish ones.
In fact according to BL Davies:
"Against highly mobile Tatar mounted archers riding in loose or no formationthe husarz echelon charge with lance was less effective. The Quarter Army cavalry would have to rely more on their pistols and carbines and attack in smaller clusters of a few poczty to try to drive the Tatars into denser formations more vulnerable to infantry and artillery fire or lance charge; and they would have to be ready to fall back to the cover of the tabor if counterattacked by too large a mass of the enemy. A tabor of wagenburg type was especially useful to the Quarter Army because it could be formed quickly in response to a surprise attack."
Warfare, State and Society on the Black Sea Steppe, 1500-1700 p.37
Note that the Crimean tartars seem to have lacked heavy shockers (and would be one of the weaker nomadic armies), which is not the case with nomadic (or other Central Eurasian) cavalries further east.
The Mongols and Jurchens were long noted to be able to charge with bullet resistant heavy cavalry working coordination with heavy lancers within a single squadron of a 50. The Polish and Lithuanian Hussars are just heavy lancers or pistolmen, whereas the Mongol and Manchus are coordinated heavy lancers and mounted archers, with each solider riding 2-4 horses, reducing fatigue and increasing mobility.
The Jurchen army at Sarhu was able to destroy the Ming army even when the later had wagons, firearms, and superior numbers.
) If nomad cavalry was actually better, give me example of battle where they actually won battle against hussars with equal numbers? They shared huge border so there should have been plenty of opportunities for such battle.
