Dennis:  Inspired by the moon, humans have employed crescent shapes for numerous symbolic purposes, for thousands of years.  Hence historical links are nebulous.  Please take a look at “Star and Crescent” in Wikipedia where symbols from diverse periods and entities are shown.   For instance, there is a Byzantine emblem from the first century BC, a Roman coin from the 2nd century,  and Crusader coins from the 12th & 13th century.   Thus, Crusaders did not view it as an Islamic symbol 
Apparently, after the Turks conquered Constantinople, they continued to employ the star and crescent symbol long employed by the Byzantine Empire.  Today the crescent appears in flags of many countries which are predominantly Muslim, and  the Red Crescent Society performs humanitarian endeavors similar to the Red Cross.