Ottoman Turkish was a language of administration and elite literature, and more or less died with the empire.

First, like all of the big early modern multi-ethnic empires (Habsburgs, Mughals, Qing) the Ottoman Empire was multi-lingual. Arabic and Persian in particular, were languages with a lot of speakers and writers and a large corpus of religious and literary works. (They were also languages that educated Ottomans would learn in their own right.) There was also a clear divide between the formal spoken and above all written forms of each of these and the daily speech of ordinary people. Again, this is quite typical.

Ottoman Turkish, according to Woodhead, was based on the “Old Anatolian Turkish” that was spoken and written in the 13th-14th centuries and remained the common idiom of Turks down to the present. She claims that the poetry of Yunus Emre (d.1310) can still be read by modern Turks, without much trouble, which is of course different from the situation in most European languages.

Ottoman Turkish was heavily influence by Persian and Arabic vocabulary and syntax. If I can risk a quote,

Mehmed Nergisi (d. 1635) said that

“the Turkish [sic] language of pleasing expression [is] distinguished by its gathering from the surrounding green meadows of various languages the choicest flowers of meaning approved by men of eloquence and, through collecting thence the fruits of clarity, admired for its natural qualities of pure and sound measure agreeable to the palate.”

Woodhead (pg 152) says that

“The Ottoman text of this passage contains around 5 per cent vocabulary of Turkish origin, 20 per cent Persian and 70 per cent Arabic; it is held together, typically, by the Persian izafet grammatical construction”

There were always people who wrote in (and championed) “Simple Turkish” but the elite language was much more complex. So while Ottoman Turkish remained a language that the elite all over the empire would want to be able to deal with, it was not a common language of day to day speech and trade, even for “Turks” (however you define that term). Still there were lots of people all over the empire who used it, either for official documents or literary purposes, and there were lots of hybrid forms, such as the Aljamiado poetry written in the Albanian language with Arabic script and lots of Turkish and Persian vocabulary, or Karamanli-Turkish written in the Greek alphabet by Turkish-speaking Greek Orthodox Christians. (Woodhead p.149)

When the empire fell, Ottoman Turkish was rejected or forgotten pretty much everywhere. This again is very typical. Modernizing states want a single national language, and almost always this involves rejecting some of the overly “ornate” and “cosmopolitan” elite language. This was particularly true in Turkey. In 1928 Ataturk decreed a new Turkish, written with the Western alphabet and with as little “foreign” influence as possible. Other modern states of the old empire were busy with their own linguistic nationalist projects, and this involved rejecting Ottoman Turkish and, of course, having no interest at all in the new modern Turkish.

That said, while Turkish is mostly a language of the modern state of Turkey, Turkic languages (including Kazakh, Uzbek and Azerbaijani) are spoken in many places in Central Asia, and the Pan-Turkic movement tries to create a common identity among all these “Turks”.

Sources

Aytürk, İlker. “Turkish Linguists against the West: The Origins of Linguistic Nationalism in Atatürk’s Turkey.” Middle Eastern Studies 40, no. 6 (2004): 1–25.

Woodhead, Christine. The Ottoman World. Routledge Worlds Ser. 2011. Chapter on Ottoman Languages

First of all, you have to note the distinction between Turkish (the language of the country Turkey and before that the Ottoman Empire), and a Turkic language. The Turkic languages are a group of at least 35 languages, like the Slavic languages, the Romance languages etc. Uzbek is a Turkic language, not a Turkish dialect.

Moving on to the Ottoman Empire. In the OE, there was this concept called "The Three Languages". Those were the three distinguished languages according to the Ottoman view. Knowing those three languages was the mark of erudition and sophistication. First was, obviously, Ottoman Turkish, which is very similar to modern Turkish aside from the script. Then there was Persian. Persian was the language of court and literature before it was edged out by Ottoman Turkish, but it was still considered an acceptable language. Last there was Arabic, which was the legal and religious language of the OE. So native speakers of Arabic had little incentive to replace it with Turkish, and that's the main reason Turkish didn't become widespread among them.

However, the Ottoman Empire had many subjects whose first language wasn't Turkish or Arabic. Let's take for example the Balkans. Several Balkan countries were in whole or in part conquered by the Ottoman Empire for several centuries. During that time, Turkish was absolutely spoken in what is now Greece, Bulgaria, Macedonia, Serbia etc. It was practically mandatory to know Turkish well if you had business with Ottoman administration, court etc. So why didn't Turkish take deeper root there?

First, because Turkish is very different from all other languages spoken on the Balkans. Turkish has a lot of cases, the sentence structure is much different, it's a difficult language if your first language is dissimilar to it. The Turkish state also never financed schools for non-Muslims. The Christian and Jewish Balkan communities financed their own schools out of donations, and naturally hired non-Turkish teachers who taught the children in their native languages.

Second, because non-Muslim Balkan women had little need to learn Turkish. The only non-Muslim Balkan woman who would need to communicate with Ottoman administration, court, government, landlords etc would be a widow with no adult male relatives. In the very rare such cases, the widow would hire a Turkish, or at least a Muslim man, to represent her in front of the Ottoman authorities. This limited the spread of Turkish further.

Third, because among non-Muslim Balkan people, Turkish (and Persian, and Arabic) were never considered prestigious languages. Turkish was the language of the hated occupants from the non-Muslim Balkan point of view. Prosperous non-Muslims on the Balkans wanted their children to learn Russian, French, or German, not Turkish. The high culture non-Muslims on the Balkans admired came from France, the Austrian Empire and/or the Russian Empire.

That said, Balkan languages were absolutely awash with Turkisms at the time those countries managed to liberate themselves from Ottoman rule. After liberation a systematic effort was started by the governments and often the intelligentsia of those countries to return the purity of Greek, Bulgarian etc, and a large number of Turkisms became archaic and are no longer in use. Even so, many, many more of those Turkisms survived, to the point modern speakers of Bulgarian etc often don't even know what word is of Turkish origin.

Other Turkisms have now evolved to mean something pretty different in modern Balkan languages than what they mean in modern Turkish. Connections are often difficult to trace. For example, in Turkish "tuz" means salt. In Bulgarian, "tuzar" is a slang word that means "rich man, vip". Salt used to be very expensive, hence a man who deals in salt = a rich man, an important man.

So, in a way, loanwords from Turkish are alive and well in some non-Turkic languages, even if Turkish as a whole language is not as popular.

Sources: History of Bulgaria volume 6, edition of the Bulgarian Academy of Sciences

Language problems in the Ottoman empire, Yelda Saydam

Османското законодателство и българското обичайно право (XV-XIX) /Ottoman Legislation And Bulgarian Customary Law (XV-XIXc), Galabina Petrova