In medieval Europe, Homer and his works were continually popular, but his texts were not available to a real reading audience until the 15th century, when the texts were "re-discovered" and translated from Greek into Latin. His renown prior to this would have been through citations and quotations in Roman rhetorical treatises (or other medieval works that cited and quoted Roman sources) that were popular choices for use in medieval classrooms, like Quintilian's Insitutio Oratoria, in which Quintilian suggests students should begin their studies “by reading Homer and Vergil" (I.8.5), even though medieval European students would only have access to Vergil. Horace, too, in Ars Poetica, says students of rhetoric should follow the advice of the Muses, and Homer is a great guide (ll.159-172).

However, I'm a little confused by the second part of your question. Homer wrote c.8th century BCE, which is about 1500 years before there was any Ottoman Empire to speak of. Do you mean the Greek speaking world? This would, of course, include the Byzantine Empire (aka Eastern Roman Empire, at least for a while).

Hope this helps a little. Happy reading!