Why would they? The Barbary corsairs had a great line in raiding the coasts of Spain and southern Italy and selling the inhabitants into slavery and Using them to man their galley fleets. The ottomans meanwhile were the beneficiaries of the great overland traderoutes from India, China and the spice islands through Persia to Syria and seaward to Cairo via the Red Sea. They were actually very worried that Portuguese exploration would damage thee trade and fought some serious engagements with them in the Indian Ocean in the 1520s (I think) which the Portuguese lost. In addition they were busy managing a large, complex and diverse multi ethnic empire and undergoing a long period of expansion (in Hungary and Egypt as well as a long and expensive war with safavid Persia) just as the age of discovery was heating up. Essentially they benefitted greatly from things as they were and had no reason to sink large amounts of capital into harebrained schemes like the colonisation of the new world.
As a further point they both used galleys which were built for the very particular climactic conditions of the Mediterranean (the basic absence of storms during half the year, allowing for very shallow ship designs among other things) and fared badly on the open ocean. This stood in contrast to Atlantic nations like Holland, England and Portugal who had always had to deal with less clement seas and as such developed naval technology to match. This is not to say that ottoman naval technology was inferior, simply adapted to the climactic conditions tha they faced.
Also, had they even wanted to the Mediterranean political environment made any expedition impossible. The Ottomans were locked in a quasi war with Venice and Spain for pretty much all of the sixteenth century of which Barbary slave taking mentioned above is a part. After the ottoman failure to take Malta in 1565 and the battle of Lepanto in 1571 the ottomans were basically confined to the eastern Mediterranean with Spain retaining control of the west and the Venetians the Adriatic. Getting beyond Gibraltar would have been a dicey prospect even had the ottomans wanted to. Keeping any such venture free from foreign interference would have been both impossible and not worth the effort.
This is a good question that has been asked before. As you seem to be exploring this topic, I recommend that you read through previous threads on this topic: