They tried. The Treaty of Sevres not only carved out large areas of the modern Republic of Turkey to give to Greece and Armenia, it also divided most of the rest of the country into French, British, and Italian spheres of influence. Shortly after the surrender of the Ottoman Empire, the country's capital of Constantinople was occupied by British and French troops, and the new government was little more than a puppet regime, as illustrated by the arrest of opposition-aligned officials by Entente forces.

Unlike in Austria-Hungary and the Arab World, the partition of Turkey was not successful owing to the different condition of the Ottoman army. While the German and Austro-Hungarian armies were routed by the time their countries agreed to an armistice, the Ottoman army and frontline were still in tact when their war ended: the Ottomans were not forced to surrender by battles, but by the surrender of their ally Bulgaria, which left their capital open to attack.

The post-war government agreed to cooperate with the allied occupation's demands for two main reasons: first, they were, in effect, hostages, and any official who sought to leave Constantinople (as Mustafa Kemal did) had to plot an escape. Second, the new government was dominated by conservative allies of Sultan Vahdettin, the enemies of the wartime Turkish nationalist regime headed by Enver Pasha. In spite of this, local commanders of the Ottoman army resisted demands to disband, and refused to hand over their weapons. In the East, general Kazim Karabekir was still waging a war against the newly formed Armenian government over the land they had been promised at Sevres.

The continued existence of Turkish military formations left the Turks in a unique place among the central powers, and, chafing under the treaty's terms, they effectively continued World War I by themselves. At first, this would seem like a fool's errand, but critically the strongest of the allied powers - Britain, France, and Italy - were exhausted by war. Britain provided only limited support for Greece, while France and Italy, distrusting Greek Prime Minister Venizelos and British Prime Minister Lloyd George, outright sabotaged their designs. This infighting among the allies allowed Mustafa Kemal, who had taken control of the Turkish army after fleeing Constantinople, to make peace with the French and Italians and concentrate his efforts on the other three powers.

Turkish forces turned the tide quickly against the outnumbered Armenians, whose entire country was occupied up by the Soviet Red Army in 1920. However, the large and well-armed Greek army initially fared well against the Turks. Greek capabilities declined following the restoration of King Constantine in 1920, and the deposition of Prime Minister Venizelos. Venizelos's allies occupied important positions throughout the military hierarchy, and their purge greatly weakened Greek organization. The Turks, meanwhile, gradually withdrew while building up their forces, which were eventually strong enough to counter-attack. After the Greeks were driven from Anatolia, Kemal threatened war against the UK if it did not evacuate Constantinople. This precipitated a crisis in London that led to British withdrawal and Lloyd George's ouster.

Turkey benefited from the premature collapse of Bulgarian forces on the Salonika front in 1918 - while the Ottoman army was not totally routed by 1918, it was steadily losing ground in Syria and Iraq and, with a few months more fighting, could possibly have collapsed. The Bulgarian surrender, while causing the collapse of Enver Pasha's government, preserved the Turkish army, which was then able to re-start the war in more favorable political conditions (after the Entente had "won" and their publics wanted peace).

Sources:

Friedman, Isaiah. British Miscalculations: The Rise of Muslim Nationalism, 1918-1925.

Travlos, Konstantinos, Akgul, Oner et al. Salvation and Catastrophe: The Greek-Turkish War, 1919–1922