Keep in mind, the Mongol Hordes conquered much larger swathes of territory than Islam or Christianity ever did, but very little of Mongol culture or religion remains in those conquered territories because Mongol culture and religion held little appeal for non-Mogols.
In the 14th century, Baghdad was probably the most advanced and civilized place in the world. However, the internal contradictions of Islam led to the overthrow of the open, advanced version of Islam present at the time and replaced it with a more fundamentalist version that is very bad at dealing with modernity.
Unless and until Islam goes through something akin to the Reformation, it is simply not able to meet the demands of a modern, advanced civilization. Adherents of literal Islam are forever doomed to live in backward societies that will constantly be passed by more dynamic cultures.