No, they didn't. The mediterranean Ottomans were using galleys in the era of carracks and caravels. Galleys are totally unsuited for traveling the Atlantic. If anthying, they could have used dhows, but those are used (to the present day) for merchant travel in the Indian ocean.
In fact, Muslims were the reason Columbus was seeking a western route to India.
Good point about the Ottomans.
The Piri Reis map is from 1513, which is 21 years after Columbus’ first voyage set out, hence there’s no basis for the claim of priority one sometimes sees. Also I think Robert Schoch’s analysis of the supposed accuracy and terrain on the map is correct.