Because the Turks were "early adopters" of hand-held firearms. They started using handgunnes in the 1440ies-1450ies, and by the early 16th century they were using matchlock arquebuses on a massive scale.
The muskets used up until circa 1640 were quite heavy and required musket rests, during the early stages of the English civil war lighter muskets which didn't require rests came into use, it took many years for the heavier muskets to die out though.
With regards to the crossbow, although it did die out gradually in 16th century Europe, the Holy Roman Empire is an exception, the Emperor Maximilian I issued an edict banning crossbows in 1517. Crossbows continued to be used for hunting up until the 18th and 19th centuries.
The long bow clung on for longer; a large amount of longbows were recovered from the Mary Rose which sank in 1545. It slowly died out in the second half of the 16th century despite attempts by the English government to keep it alive. The main reason being the lack of men willing to take the long and arduous training required to use it.
In the East bows continued to be used for much longer. Turkish sipahis, Polish pancerni cavalry, and Russian cavalry continued to use composite bows until the late 17th century. In India, Central Asia and the Middle-East bows continued to be used by horsearchers until the late 18th century.