Eh, so did Christians.
What really crippled the Byzantines is when the CRUSADERS trashed and plundered Constantinople earlier. Then they were easy pickings for the Turks later, who finished them off.
Yep. 4th Crusade, one of the most interesting watersheds between the medieval and the modern. The biggest problem the Greeks had was that the Catholics in Rome were trying to get them to convert and they didn't want to, hence were very tardy in hoping for military help from a country that had plundered them three hundred years before in similar circumstances.
In fact, they had, to that point, better success coexisting with the Muslims than they did with their nominal co-religionists, so it seemed more of a tossup then as to which side was worse news for them than it might seem today. The Italians finally did send help, too little too late, and with theological strings attached. IMHO, were it not for the schism in Christendom a city that had been Christian from its very founding might still be so.
This date was one Gibbon used as his own annus mirabilus - the 100 Years' War ended this year and Gutenberg was just starting to use his new printing press. It's hard to believe how much of the way the world is today was set in place at this time.