It's difficult to explain why something didn't happen, but I will take a stab at it. I think the two reasons are related to the areas that the Turks had conquered and the process and timing of decolonization.

I'm going to ignore areas that the Turks held briefly and not into the 19th century.

Once the Turks had settled in Central Anatolia and mostly united under Ottomans, they began to conquer territories in the Balkans populated by Greek speakers, and speakers of a variety of related Southern Slavic languages/dialects. They later conquered areas that spoke the predecessor of Romanian. None of these languages have much in common with Turkish.

In the early 16th century the Ottomans conquered the Mamelukes in Egypt and came to possess Egypt as well as most of the Middle East. This area primarily spoke Arabic.

Many territories controlled by the Ottomans were controlled for at least a period of time as locally self-governing vassals, which would generally have functioned in their local language. Even within the core Ottoman territories, different religious groups had their own form of self government, which would have functioned in their native languages (mostly Greek) rather than Turkish. Arabic retained its status as the primary language of Islam, and also the first language of the vast majority of people from Morocco to the Persian Gulf.

The process of independence for the European subject peoples in the Ottoman Empire was deeply intertwined with the concept of Romantic Nationalism, the idea that each nationality had a unique culture tied to language and folk history, and that these nationalities deserved their own states. This process unfolded between 1821-1912 and to a relatively large extent (at least a much larger extent that what occurred in Africa) led to states that were relatively linguistically homogeneous, or at least had one clearly dominant language that the state had an interest in promoting.

During the 19th and early 20th centuries, these states were generally pretty hostile to the Ottomans/Turkey, and were integrating themselves into the European alliance systems. Countries like France, Germany, and Russia were competing for alliances with these new countries, and both because of that and the linguistic similarities that existed in many cases (Slavic languages with Russian and Romanian with French), Balkan residents were much more likely to end up using a European language as a second language than Turkish.

The Arab possessions also became independent by the end of WW1, and I think the answer here is much simpler. Arabic was a dominant language with prestige connected to Islam, and there was no reason for anybody to need an alternative lingua franca in the Greater Middle East.

These factors are all in significant contrast to Sub-Saharan Africa for example, where state boundaries were drawn with no regard for language, and where a variety of languages even in relatively small geographical areas necessitated a lingua franca.