In A Shameful Act: The Armenian Genocide and the Question of Turkish Responsibility (2006), Taner Akçam describes an interesting process. He claims that the mass killings were not denied by the Turkish public; instead, this process emerged after 1920 when international agreements seemed to suggest the creation of an Armenian state in six provinces in the east of Turkey, while the Greek army threatened to reduce Turkey's area by taking some of its western provinces.

The same treaty, the treaty of Sèvres, that threatened to reduce Turkey's size also stipulated that trials be held against perpetrators. It is easy to see, thus, why Mustafa Kemal and the rest of the elite linked the borders of their newly-founded nation-state, already a truncated form of the former empire, and the responsibility taken or denied for mass killings. This had led to an operation in which the National Movement reached out to members of the CUP and others in an attempt to rationalize their actions, then making sure they can be forgotten.

The topic of the Armenian Genocide (ethnic cleansing of all Armenians from eastern Anatolia/western Armenia) became a nationalist issue in Turkey during the 1920s and 1930s as u/Orel_Beilinson described in his answer, citing from Taner Akcam's book, it was because of territory loss and this paranoia arising from the Treaty of Sevres that the West and Armenia had conspired to carve up Turkey.

Sevres gave a large chunk of Ottoman territory to Armenia in the east (Van, Erzurum etc.), and the borders of Wilsonian Armenia were enlarged to compensate the Armenians for their suffering and to punish Turkey. In Sevres the Ottomans accepted responsibility for massacres of Armenians (ARTICLE 142), promised to prosecute their war criminals (ARTICLE 230), and concede territory to Armenia in return (ARTICLES 89-93). However what this did in Turkey was to associate Genocide recognition with land-loss and national surrender/humiliation, and this kept the Ottoman narrative alive that the Armenians were pro-Russian separatist who were disloyal and rebelled against them during WW1 for land-grabs (rather than self-defense as Armenians claim).

This explains why many nationalist Turks are so defensive about this topic over a century later. If you recognize the genocide and do not adhere to the official Republic of Turkey historiography, you are basically viewed as a sell-out/traitor by some.

In the end, Turkey under Mustafa Kemal won their war in rejection of Sevres (1919-1923) and took more land from Armenia in their 1920 invasion and expanded their borders into the Caucasus re-taking land the Ottomans had lost to the Russian Empire in the 1870s. This was a huge success for Turkey. So even though Turkey had won and no core territory was lost, nonetheless there still remained this nationalist paranoia and association of Genocide recognition with land-loss.

I'll add that Mustafa Kemal, while accepting the Armenians were treated unjustly, realized the Armenian Genocide had become a nationalist issue and early on he took steps to write his own narrative of history for the new Republic of Turkey. Kemal did this by dismantling Sevres one by one, which to Turkish nationalists was a national humiliation and conspiracy to destroy Turkey.

For example, Kemal secured the release of the Ottoman war criminals held in Malta in 1920 and pardoned them (discarding ARTICLE 230) and he invaded the short lived Republic of Armenia and land-grabbed (discarding ARTICLES 89-93 that called for recognition of Armenia), and lastly he rejected the Western narrative of the Armenian genocide.

So while the Republic of Turkey under Kemal in its early days shunned Ottoman legacy and embraced the Hittites, it was very nationalist and the topic of the Armenian genocide was associated with land-loss. That is why to this day, there is the perception Turkey recognizing the genocide would somehow compel them to concede land to Armenia or make them guilty in the eyes of the world for having the land that was ethnic cleansed of Armenians. So they've dug-in and are holding on their narrative.