Reading your question, I feel that with fascism, you mean something like an irredentist and aggressive foreign policy. Is this interpretation correct? I will move on assuming it is so but do tell me if you meant something else.

To begin with, the Turks had something Germans, Hungarians and all other 'defeated nations' did not. Germans submitted to the Treaty of Versailles. Though Weimar government officials continuously sought to flex the boundaries introduced by Versailles, with some success, it was the Nazis who completely rejected it. Other defeated nations had their own treaties which severely limited their activity. The Ottoman Empire was not an exception; they signed the Treaty of Sevres in August 1920. This treaty gave large parts of the former Ottoman land to Armenians, Greeks, to French and British mandates, to Italians, even some to Georgians. As u/Iphikrates points out, it is very difficult to prove the counterfactual in history. But we can guess that had this treaty been implemented, Turkey (or whatever remained of it) would have probably joined the Axis. Yet, as I said in the beginning, Turks had something other defeated nations did not have and that was a nationalist resistance movement to the Allies. Immediately following the Allied invasion of Turkish cities, spontaneous and local resistance movements emerged in parts of Turkey. Mustafa Kemal Pasha (later Atatürk) organised these groups, unified them, and helped create a new, alternative parliament in Ankara independent from that in Istanbul under indirect (and later, direct) British rule. First, the Italians left. Having many problems already at home and finding themselves at odds with the Allies on other issues, facing a Turkish resistance seemed unnecessary for them. Then, the remnants of the Eastern sections of the Ottoman Army refused to lay down arms, joined forces with the Ankara government and forced the Armenians out of Ottoman borders in late 1920. Independents groups led a rather effective guerilla warfare campaign against the French in southern cities which led the French to realise their interests were better served by recognising the Ankara government in 1921. Finally, a newly founded regular army of Turkish nationalists led by Mustafa Kemal managed to defeat the Greek Army (materially and politically supported by the British) decisively in 1922. The Allies recognised their defeat in 1923 and agreed to a revised treaty (Treaty of Lausanne) which saw Turkey get back much of the lost Turkish-majority lands lost in Sevres. So the Italians had what they called Vittoria mutilata (mutilated victory for the results of the War were not what they saw fitting for themselves as a victorious nation) and the Germans had the Treaty of Versailles. The Turks had the War of Independence. That was a key difference.

Following the War of Independence, there were still problems between Turkey and the Allies, of course. Hatay (the Sandjak of Alexandretta back then, now a Turkish province famous for its delicious food!) was one such problem as Turkey sought to recover it back from Syria under the French mandate. They got it back through peaceful methods and negotiation in 1939, ironically, in part, thanks to the necessity of the French to relocate troops from Syria to deal with the irredentist threat in Europe.

Furthermore, one must also consider, I think, the impact of a continuous series of wars that involved the Ottoman Empire from the Italo-Turkish War in Tripoli (modern-day Libya) in 1911 through the Balkan Wars in 1912-13 to the First World War. The Ottomans emerged defeated from nearly all of these wars, lost many people, and had to deal with an immensely failed economy. Together with this, the new government under Atatürk eyed radical, Western-inspired reform to transform the country from being a 'backwaters Eastern country' to a 'modern civilised country', as they would put it themselves. The Kemalists reasoned that engaging in warfare did not prove conducive to such reform attempts. Some of them were personally suspicious of 'empire' and imperialism of all sorts; Atatürk himself once compared the Ottoman sultans to Napoleon Bonaparte and stated these ruined their nations in pursuit of imperial dreams (Atatürk'ün Söylev ve Demeçleri, vol II, pp. 155-56). We have reason to think this mindset dominated Kemalist circles and hence they sought to create a web of regional alliances and non-aggression pacts (including, initially with Italy and Nazi Germany) to avoid conflict in Turkey's region.

Sources and Recommendations:

Atatürk'ün Söylev ve Demeçleri, vol. II (Ankara: 1997).

Bernard Lewis, The Emergence of Modern Turkey (London: 1961).

Feroz Ahmad, The Making of Modern Turkey (London and New York: 1993). I would especially pp. 66-68 where Professor Ahmad summarises the Turkish perceptions of Italian fascism and aggressive foreign policy followed by Italians and others in the 1930s.

Eric-Jan Zürcher, Turkey: A Modern History (London: 1993).

Hey there,

Just to let you know, your question is fine, and we're letting it stand. However, you should be aware that questions framed as 'Why didn't X do Y' relatively often don't get an answer that meets our standards (in our experience as moderators). There are a few reasons for this. Firstly, it often can be difficult to prove the counterfactual: historians know much more about what happened than what might have happened. Secondly, 'why didn't X do Y' questions are sometimes phrased in an ahistorical way. It's worth remembering that people in the past couldn't see into the future, and they generally didn't have all the information we now have about their situations; things that look obvious now didn't necessarily look that way at the time.

If you end up not getting a response after a day or two, consider asking a new question focusing instead on why what happened did happen (rather than why what didn't happen didn't happen) - this kind of question is more likely to get a response in our experience. Hope this helps!