Why would they want to, even if they could? While this a counter-factual situation, the Ottoman Empire not long after 1492 reached its maximum territorial extent that covered up to East-Central Europe, the Arabian Peninsula, North Africa, and up into the Black Sea region with their vassal state the Crimean Tatars. The Ottomans effectively had two active war fronts between the Habsburgs in the West and Persia, who they considered their most important rival. To put it in perspective, during the Classical Era under Selim I and Suleiman the empire was tied up fighting wars on either front for a great deal of time during the latter's reign. For another bit of perspective, Selim I conquered the Mamluk state in 1517. Spain conquered Mexico just a couple years later from 1519-1521. Ottoman strategic priorities in this case were tied up already, and they already possessed a favorable geographic position in terms the emerging global economy. In addition, the Mediterranean was a contested zone between the Ottomans and various European states even in the period when the empire had near-undisuputed naval dominance. Coordinating logistics between the New World and the eastern Mediterranean would have been difficult to say the least.
The Ottomans did take part in what we would call the Age of Exploration in other ways. There was limited interest in the New World by the Ottomans evidenced by at least one manuscript consisting of pseudo-translations of European texts about North and South America. But there doesn't appear to be documentary evidence of an Ottoman subject even travelling to the New World (as far as I know at least). In the Classical Era there were also Ottoman naval expeditions into the Indian Ocean and potential discussions of a Suez Canal that did not come to fruition. The empire also competed in this region with the Portuguese and fought land and sea engagements with them. Overall though, the short answer is that the empire simply had too many priorities in their immediate spheres of influence and not enough commercial motivation to colonize or trade in the New World. I would be interested if some day there was a source discovered from an Ottoman subject who experienced the New World first hand, but unfortunately that doesn't seem to be the case.
Some additional reading:
Giancarlo Casale, The Ottoman Age of Exploration
Thomas D. Goodrich, The Ottoman Turks and the New World: A Study of "Tarih-i Hind-i Garbi" and Sixteenth-Century Ottoman Americana
Hey there,
Just to let you know, your question is fine, and we're letting it stand. However, you should be aware that questions framed as 'Why didn't X do Y' relatively often don't get an answer that meets our standards (in our experience as moderators). There are a few reasons for this. Firstly, it often can be difficult to prove the counterfactual: historians know much more about what happened than what might have happened. Secondly, 'why didn't X do Y' questions are sometimes phrased in an ahistorical way. It's worth remembering that people in the past couldn't see into the future, and they generally didn't have all the information we now have about their situations; things that look obvious now didn't necessarily look that way at the time.
If you end up not getting a response after a day or two, consider asking a new question focusing instead on why what happened did happen (rather than why what didn't happen didn't happen) - this kind of question is more likely to get a response in our experience. Hope this helps!