I thought of something that seems unlikely, but I figured I could share anyway. What if an Ottoman Sultan or prince of a smaller Ottoman Empire, feels as if he is shunned by Allah and converts to Catholicism or Orthodoxy. He travels around the Mediterranean and learns the languages. He wants to act like a western European nation, so he makes Latin the language of the court, and forces his lords to speak latin. Latin eventually merges with Turkish over the coming decades or potentially centuries if this policy is kept. Turkish, or at least a branch of it, becomes romance and is spoken in the smaller Ottoman Empire.
I'm no expert of Ottoman law, but I'm guessing they wouldn't let him either stay in or get power if he converted.
Exactly. The Sultan's legitimacy came from Islamic Law, and if said sultan ceases to be a Muslim... well, Sultans have been deposed for a lot less. And such a language would be classified as Turkic anyway. Turkic with a lot of loanwords from Latin, but still Turkic, just like how English is Germanic and not Romance.
By the time you have Turks in Anatolia it's probably too late. Even the Mongols, the strongest steppe confederation in history, didn't take Italy or France - too many other strong states in the way, too far from the steppe before civil war set in, too many peoples who'd have their customs absorbed first even if the empire did get that far. Heck, by the time you have the ethnogenesis that leads to the modern Turkic subgroup called "Turks" it's probably too late - too many Muslims in the way.
The closest I can come is to get a Turkic group involved in the migrations period, become the ruling class, and eventually assimilate to the language of their subjects - like the Franks did in OTL France. And OTL, there may have been Turkic peoples involved. The Huns may or may not have been Turkic, and if they were (which is an if - we know very, very little of their language) then a bit more luck could see Atilla live longer and his heirs found some kind of lasting state on conquered Roman territory, albeit probably not holding onto all the lands of his short-lived empire.
The Eurasian Avars are securely Turkic, and although OTL their empire was largely Slavicized, not Romancized, before its fall, perhaps a bit more luck in war would see them actually take a significant chunk of Italy instead of trying and failing; if so, they might absorb Italian customs and language, and the name would survive as a region in Italy where many of them settled ala Lombardy (from the Lombards, a Germanic people) OTL.