The Pope's crusade? He excommunicated them prior to the attack.
Why not look in the mirror to your own Roman Emperor, Alexius IV Angelus, who paid them off to attack the City?
And given the record at Mazinkert and afterwards, and the shoddy treatment of the Emperor by the Romans then, as well as events like the pointless Civil War from 1341-1347, I doubt the East Roman Empire would have lasted to 1571 anyway.
The 4th Crusade did not hasten a Turkish takeover. Fratricidal Civil wars and 14th century warfare between the Serbs and Greeks was sufficient for that, and to allow the Paleologeans to lose all of Asia Minor.
Get off your high horse.