I might be completely incorrect, but IIRC a lot of the modern area wasn’t under Turkish rule until much later when they conquered and largely ethnicially cleansed the area to repopulate it with Turkish people. Hence, which people were the natives to this area, before the turks (and the ottoman empire)?
You aren't completely incorrect, but you are conflating a couple different events, namely the Turkic migration into Anatolia in the first centuries of the second millennium CE (this is what led to the "Turkification" of Anatolia) and the population transfers and genocides of the early twentieth century. For the former, while the conquest and settling of modern Turkey by Turkic peoples was of immense cultural and political significance (one illustrative example being a sharp decrease in olive cultivation) it was of only somewhat minor genetic significance: modern studies indicate about 20% of modern Turkey's genetic makeup is of Eastern Asian origin (crucially, this does not mean that in the eleventh to fourteenth or so centuries a fifth of Anatolia's population was replaced, the region has been a "heart of the world" for a very long time and had seen significant migration and immigration because of that). In practical terms, what this means is that the ancestry of people who live in Turkey today is overwhelmingly the people who lived there in past mellenia. The process of "Turkification" was one of assimilation, not replacement.
For the genocides and population transfers of the twentieth century (and I use the term "population transfer" advisedly, because frequently the ethnic cleansing of religiously defined minority populations from modern Turkey was accompanied by simultaneous ethnic cleansings whereby "Turks" from nearby regions were forced into Turkey in rather less well known ethnic cleansings) while this had major local effects it would not be enough to constitute a "replacement" throughout Turkey as a whole. Furthermore, the groups inflicting genocide or having it inflicted upon them were defined religiously or linguistically, not genetically.
That's a loaded question since Turkey being transcontinental was the subject of many migrations and conquests. The answer depends on what you will ultimately consider "native" since it has a lot of meanings.
If we go by one of the Cambridge Dictionary definitions, we get - native: relating to the first people to live in an area
The first people to settle modern day Turkey were Anatolian farmers, a cline of people from the initial migrations of homo sapiens out of Africa. It is these AHG (anatolian hunter gatherers) who later on migrated into Southern Europe in the peopling of Europe. Till this day, a huge chunk of mediterranean ancestry comes from the AHG. So, the first modern men in modern-day turkey would be Anatolians.
Now, modern day Turkey is a multiracial place with a population having many different ancestries. I use "racial" here with some liberty, since it doesn't have a lot of anthropological sense.
Now, "Turkic" people related to the Seljuk Turks (and Ottomans) are the ones responsible for the modern day culture, language and tradition of Turkey. They completed the process of turkification of anatolia and thrace from hellenic cultures. However, their genetic impact was very minor due to their lower population. They were however very successful in completing an elite transfer of Anatolian culture (putting themselves at the top of the social hierarchy)
So modern day "turks" are a mix of anatolian farmers, mainland europeans (WHG & Steppe) and Turkic tribes (with the original turks originating in manchuria and migrating west). A lot of the turks in thrace are very recent transplants from the balkans due to Ottomans and hence further add to a recently constructed modern turkish identity that is based on a common culture, language and ethos.
To conclude, most of the modern day Turks get majority of their ancestry from AHG and some
of it from continental europeans so they can in theory consider themselves the natives of the land. However, modern turkish identity is somewhat like modern american identity. Just because you look a certain way will not mean you're not considered turkish if you were born and raised there + follow the culture, speak the language and adhere to the religion.
Sources: Genome Sequencing of Turkish Populations
Pleistocene migrations into Anatolia
Essays in Ottoman and Turkish History; Roderic Davison