But, was it rising faster than Islam at that time? It wasn't until missionaries started converting folks in the "New World" and parts of Asia that Christianity really rose.
When Constantinople fell, Christianity was not close to being on the brink of extinction.
Western European Christianity had finished converting millions in the Europe and Russia, through force (the northern crusades), and the Reconquista was a massive success by that time.
You also need to keep in mind that while the Ottoman Turks were doing well at that period, the Caliphate had for all intents and purposes fallen to peices, so muslim nations were also having significant problems.
In Europe Christianity had rose from 400 or so on. They didn't convert the new world because they had not discovered it, but had been making significant inroads into previously pagan areas throughout the middle ages.
I don't think you could find a serious historian who would argue that when the small remaining stump of the eastern empire fell to the Ottomans, christianity was anywhere near extinction. Especially when you consider how tolerant the Ottoman empire was towards Christians and Jews.