While built on three continents (Asia, Africa and Europe) and on the legacy of two empires (Byzantine and Seljuk), the Ottoman Empire [OE] was very successful in managing its demographics in an orderly and efficient manner simply by promoting cultural co-existence and providing relative freedom to be who you want to be. Society was stratified and there was no cross-strata mobility but everybody was equally protected. For many people everyday life didn't change, only taxes and social duties did. The state institutions were just a way to organise everyday life, not to change it. The Islamic law separated the Muslim state from the non-Muslim population. You could be judged by a different court, be taxed differently, and have different social opportunities depending which side of the divide you chose to be. For 500 years there was no "citizenship" but everybody within the Empire was a subject. A subject of the OE could keep their customs, with certain limitations, and Muslims enjoyed more respect and prosperity. But if preserving your own culture was a priority, you could still have a humble but orderly existence within your own people. People could mix but there was no necessity to combine. Different groups could keep their identity although choice was very much guided by peer pressure and there was a strong "us" vs "them" mentality. Joining "them" meant losing your own.

Diversity was what kept the OE together for so long and in such a successful manner. Being flexible and adaptable, and allowing what we now call "multiculturalism" helped a lot. Unlike the Spanish or the British, the Ottomans did not arrive with the claim of being the more advanced civilization nor did they try to spread their religion using missionaries. The most distinctive feature of the OE was the importance of religious identity over cultural or ethnic identity. But that importance had administrative function and served as a way to organise society. And if today you go around the countries whose territories were once part of the OE, you'll notice many cultural similarities. For example the Balkan countries have very similar traditional architecture and you'll get similar dishes in their cuisine, all legacy of the OE, including a lot of words borrowed from Turkish. But they all would be currently considered part of their own identity because the cultural assimilation would have happened naturally.

OE's tolerance and acceptance of pluralism could be traced back to the idea that Turkic rulers descended from god (according to the pre-Islamic Central Asian tradition), and the divine association made them a "world emperor". The Seljuks and later the Ottomans used the old Central Asian title "Han" as a symbol of a legitimate effort to create a world empire. One of the most famous Ottoman rulers, Suleiman the Magnificent, referred to himself as "Sultan Suleyman Han bin Selim Han bin Bayezid Han, the Padisah" where "padisah" comes from the Persian tradition, while the adaptation of the Islamic tradition brought the Islamic title "Sultan". Mixing cultures was fine. Non-Muslim communities were not assimilated.

Later on, the plural system proved ineffective as nationalism grew within some groups and independent movements disturbed the unity of the Empire but for a long while it worked quite well. Only in 1839 there was an attempt at creating a common Ottoman identity with the Imperial Edict of Gulhane. It introduced guarantees for the life, honor and property, equal taxation, and equal military conscription for all Ottoman subjects. Citizens were no longer categorized according to their religious affiliation, civil service could now be entered by non-Muslims etc. The reforms were an attempt to create a national Ottoman identity but the old system was not abolished and the legal system still allowed court separation by religion. At the same time parts of the OE had already become autonomous or had strong autonomous movements. Nationalism was already on the rise and the various groups were quite distinct and were now drifting apart. The particularization eventually led to the dissolution of the Ottoman society and the emergence of several nation-states out of the Empire.

Further reading: H. Inalcik "Ottoman Archival Materials on Millets", B.Braude and B.Lewis"Christians and Jews in the Ottoman Empire: the functioning of a plural society" , K.Barkey "Empire of Difference", F. Gocek "The Social Construction of an Empire"

As far as language goes, I suppose it depends on what you mean by "Turkish."

There was (for lack of a better name) common Turkish, which would be the everyday langage of common people. This was spoken by ethnic Turks, but was also used as a second language by non-Turks in places where they came into enough contact with ethnic Turks on a regular basis. At the end of the day, you spoke the language of your ethnic group, and maybe others, especially if you had enough contact with other groups.

But if by "Turkish" you mean Ottoman Turkish, it's a bit different. Ottoman Turkish is a hybrid confection of Turkish, Persian, and Arabic, and it took several years to learn it properly. Its source languages are all from completely unrelated language families and was written in the Arabic script (which was actually a really bad fit for it.) Knowledge of Ottoman Turkish was a sign of education and culture and was required for positions in the court or administration.

So while an educated, a thin social layer of Ottoman-speaking administrators extended across the empire, it was not a language that ordinary people needed, nor did they really have access to in terms of the time and effort to learn it.

That said, there are places in the former Ottoman Empire where Turkish has been spoken continuously to the present (by larger or smaller groups), in places such as Bulgaria, Cyprus, and Greece. However, these speakers overwhelmingly identify as ethnically Turkish as well. I can't speak to whether this is because they were Turks who emigrated from Anatolia or were locals who assimilated culturally, but in any case, these are examples of places where the Turkish language did in fact spread.

So why wasn't Ottoman Turkish more widespread given the size of the empire? By the late 19th and early 20th century, the Ottomans realized that they needed to create a public school system in the Ottoman Empire to catch up with other powers. They belatedly started opening schools, but were hindered by lack of funds. Ottoman Turkish, was taught in them because it was the prestigious form of the language; common Turkish was seen as a sort of street language and therefore was unnecessary to include in the curriculum. The number of schools and graduates were few and tended to be in the big cities, so it's not like the state was opening Ottoman-teaching schools in rural Libya or Bosnian villages.