The shift was the result of deliberate expulsion/exchange, and the flood of refugees fleeing ethnic/ religious violence.
Probably the best- and most infamous- example of deliberate population exchange followed the Greco-Turkish war of 1919-23. The 1923 Treaty of Lusanne provided for the exchange of the respective minorities in Greece and the newly established Republic of Turkey following the cessation of major hostilities, which had only occurred after widespread atrocities and massacres which culminated in the Great Fire of Smyrna. This saw almost the entire Greek minority in Turkey, with the notable exception of the Greek community in Constantinople, deported to ports in Greece, where very little preparation or infrastructure had been made ready for their arrival apart from meagre plots of land from which they could do little more than subsistence farming; approximately 1.2 million people suffered this fate. The same happened to the approximately 400,000 Turks living within Greece's borders, which had expanded enormously since 1913 as a result of the Balkans Wars and World War One. The Thracian Turks were, like the Constantinople Greeks, exempted from this transfer. Why either group were exempted is beyond me, and hopefully someone better qualified can answer than question.
Expulsion was another major contributor. Many people are aware of Stalin's deportation of the Crimean Tatars and their systematic replacement by Russians during and in the aftermath of WW2, but far fewer are aware of the Tsarist Russian policy towards them, which was virtually the same. Beginning with the Russian annexation of the Crimea in the 1783, the Tsarist authorities instituted a policy of encouraging immigration from other parts of Europe; this became so diverse that, during the Crimean War, it was noted that there were German, Italian and Greek communities all living in the peninsula. This soon led to problems, however. The Russian conquest of the Caucuses was slowed due to the skill and tenacity of its Muslim inhabitants, especially under the leadership of Imam Shamil, and this soon led to Russian fears. These came to a head during the Crimean War when the Tatars, systematically oppressed and pushed out of the best land and all positions of political authority by the Russians, joined the Allied cause en-masse and actively assisted them. The Treaty of Paris, signed in 1856, did provide for their protection, but the allies showed very little interest in enforcing this provision. Instead the Russians began the deportation of entire Tatar villages on the pretext that, during the war, some of the inhabitants had been absent without authority from their homes and surrounding area. Whilst not as widespread as the later Stalinist deportations, these were still by no means inconsequential.
Finally, we have refugees fleeing conflict. The 19th and early 20th centuries were a bloody time in South-Eastern Europe and the Balkans, and this inevitably entailed many atrocities. The Caucasian war, referenced above, was no exception, and featured intense bloodletting on both sides. Probably the most horrifying examples, however, come during the Balkan wars. Misha Glenny, in his book "The Balkans 1804-2012: Nationalism, War and the Great Powers" quotes a Russian witness to the Turkish atrocities into Thrace during the First Balkan War, who wrote that
!"The church is completely blackened by fire. On the path dead children, little girls with split skulls and legs spread apart. The criminals probably caught the crowd as they were fleeing for sanctuary. The inside of the church is a terrible stage.... once my eyes get used to the dark, i see the pile of burnt bodies of people desperate to break down the doors, locked from the outside, as the building went up in flames." (Glenny, 230-31)!<
It was not, of course, just the Turks who committed these atrocities. It was, however, the Turks and Muslims who made up the vast majority of refugees, since they fell on the loosing side. During the Greco-Bulgarian advance on Salonika, for instance, almost 20,000 were put up in Salonikan mosques in terrible conditions. The Serbs and Montenegrins began a campaign against the Albanians in an attempt to subdue them, so that they could finally win their coveted Adriatic seaport; Catholic Albanians who had initially sided with them quickly recoiled in disgust at the actions they bore witness to. Naturally, anyone who could get away did, and the safest place to flee was the rump of the Ottoman Empire; Sean McMeekin, in Ottoman Endgame, states 400,000 Muslims had fled to the remainder of the Empire by the end of the two Balkan Wars, whilst almost 200,000 Greeks had fled in the opposite direction [McMeekin, page 80]. Indeed, the influx of refugees from the Balkans would add to the ferocity of the Greco-Turkish war of 1919-23, which was seen almost as an opportunity to settle old scores.
In conclusion, there were several reasons why the Turkish/Muslim population of the Balkans fell so dramatically as the Empire crumbled. Initially, it was Nationalism in its simplest sense- space was made for the victorious state's own people, and precious little thought was given to the defeated Muslims. Many others were simply fleeing the horrors of war. Finally, and most ominously, the legalised, regulated population exchange of 1923 saw the final demographic shift, but also provided a twisted basis for the later examples of this in 1945.
I hope this helps answer your question.
Sources:
The Balkans 1804-2012: Nationalism, War and the Great Powers, by Misha Glenny
The Ottoman Endgame: War, Revolution, and the Making of the Modern Middle East, 1878-1923, by Sean McMeekin
Crimea, by Orlando Figes