As a Turkish citizen I have some answers.

We have to discuss the first migration of Turks to Anatolia and the last migration of Greek from Anatolia. Let's begin with the population exchange between Greece and Turkey.

After WWI, the partition of the Ottoman Empire by Allied Powers occurred.

So as a response with the leadership of Mustafa Kemal Atatürk (founder of Modern Turkey), The Turkish War of Independence (19 May 1919 – 24 July 1923) was fought between the Turkish National Movement and Greek in the West, Armenian on the East, France on the South, royalists and the separatists in various cities, and the United Kingdom and Italy in Istanbul.

The major battle was Greco-Turkish War or as Greek say Asia Minor Disaster. The Greco-Turkish War was fought between Greece and the Turkish National Movement, between May 1919 and October 1922.

Greece lost. As a result, the Greek government accepted the demands of the Turkish National Movement and returned to its pre-war borders. The Treaty of Lausanne recognized the independence of the Republic of Turkey. Adding that The Greek and Turkish governments agreed to engage in a population exchange.

According to the population exchange treaty, Greek orthodox citizens of Turkey and Turkish and Greek Muslim citizens residing in Greece were subjected to the population exchange between these two countries. Approximately 1,500,000 Orthodox Christians, being ethnic Greeks and ethnic Turks from Turkey and about 500,000 Turks and Greek Muslims from Greece were uprooted from their homelands. *1

There is a perfect movie Rembetiko (1983), about the migration of Greek musicians from Smyrna, Turkey. This musical drama sweeps through a turbulent 40 years in popular singer Marika's life starting with the singer's birth in Smyrna, Turkey in 1917. Then she was deported to Greece along with all the other Greeks in Smyrna when she was seven years old. Movie focuses on the difficulties of Anatolian Greeks migration with perfect Greco-Turkish songs, that makes you feel what they felt.

From that time to now tensions were always high among the settled minorities and the Turks. Therefore, minority Greeks continued to migrate for years.

So let’s talk about the migration of Turks to Anatolia.

Before Turkish migrations, Anatolia was home to many different people who were either natives or settlers and invaders. From Ancient times to Ottoman Empire including Byzantium Empire there were Armenians, Native Anatolians, Persians, Hurrians, Cimmerians, Galatians, Colchians, Iberians, Arameans, Assyrians, Corduenes, Albanians, Circassians, Georgians, Greeks, Jews, Romani, Slavs, Kurds and other minorities. So Byzantium was not just consisted of Greek.

Of course you are somehow right. By the 5th century the native people of Asia Minor were entirely Greek in their language and Christian in religion. *2

Because Alp Arslan won the Battle of Manzikert, Anatolia started to be settled by Turkish nomads. First they ruled Seljukid and then Ottoman Empires that were in significant power to impose Turkish Language and Islam to the territory.

But how much Turkish are Anatolian Turkish? That can only be understood from genetic studies. It can be thought that modern Turkish are direct descendants of the Seljuk and Ottoman hordes who surged into Anatolia from Central Asia. There have been many wars, many invasions, many migrations, many trades, namely many human interactions in Anatolia throughout history, before and after Turkification. A report in the journal Annals of Human Genetics in 2012 indicated the paternal ancestry of those living in Turkey was 38 percent European, 35 percent Middle Eastern, 18 percent South Asian and only 9 percent Central Asian. From this statistics, Turks have Greek (European) blood and of course Greek have Turkish blood. Yes millions of Greek has gone but our music, cuisine, traditions are still really alike.

1 Twice A Stranger: How Mass Expulsion Forged Modern Greece and Turkey, by Clarke, Bruce (2006), Cambridge, Massachusetts: Harvard University Press. ISBN 9780674023680.

2 Swain, Simon; Adams, J. Maxwell; Janse, Mark (2002). Bilingualism in Ancient Society: Language Contact and the Written Word. Oxford [Oxfordshire]: Oxford University Press. pp. 246–266. ISBN 0-19-924506-1.