Specifically around the question about enclaves of Turkish-speaking people: in fact there are. They are generally known as "Turkomans" or "Turkmen", yet despite the name share close linguistic and ancestral links with Anatolian Turks, rather than with Turkmen from near the Caspian Sea.

There are notable communities of Turkmen in Syria and Iraq, where they make up an estimated 1% and 3% of the population respectively.

The Turkish government in recent conflicts in Syria and Iraq has taken an active interest in organizing and supporting Turkmen groups.

There's a lot more to their history, especially from the Ottoman period, but I will leave that for an expert who is more well-versed in the region and its Ottoman period history. Just thought I'd mention that these populations did and still do exist in the region.

Let me turn the question around and begin by looking at places with enclaves of Turks. The vast majority of these places are in the Balkans: Greece, Bulgaria, Macedonia, some in Serbia and Kosovo too. These places all came under Ottoman rule between the 14th and 15th centuries. Incidentally, this corresponds to the increase of Ottoman rule in Anatolia. Prior to this, there were many independent Anatolian Principalities. When Ottomans extended their rule over these principalities through war or diplomacy, they had to deal with the nomadic Turkish population of these regions. The Ottoman state would prefer these chaps to sit where they were and pay taxes on a regular basis, but they obviously preferred rebelling, raiding, and occasionally killing the poor officials who came to collect taxes. Hence, the Ottomans forcibly relocated some of these Turkish groups to the Balkans. These would settle in, say, Bulgaria, start farming, and as their links to their 'tribesmen' would be severed, at any rate, they would be much less dangerous to the central authority. Further, the Ottomans would have a reliable Muslim population in such regions if the process went alright; a bastion against potential problems with the local Christians (which, often, did not happen at all).

Now, the population of the countries you have mentioned above were already Muslim. And by the time they became part of the Ottoman Empire (the 16th century), the situation in Anatolia had changed as well. So you do not find large groups of Turks in these countries (contrary to Iraq and Syria, for instance, which already had sizeable Turkish groups from the time of Seljuks onwards).

Second, until late in the 20th century, the Ottoman state did not aim to assimilate or Turkify its population. Nations are not as novel as some historians and social scientists assume, and it is beyond doubt that the Ottoman elites who were of Turkish origin had an idea about what being a Turk meant. Yet, the understanding of nationalism differs quite starkly from age to age. The 20th-century tendency of solidifying a country's internal cohesion by assimilating minorities did not seem to have crossed the minds of Ottomans for the majority of their history. The Greek conversion to Islam and 'Turkishness', of which we still know too little, during and after the Conquest of Anatolia by the Seljuk Turks seems to have been an automatic process rather than a conscious effort by Turks to convert Greeks. The same applies to Albanian Muslims, Bosnians, and Pomaks in the Balkans. And even these do not speak Turkish, despite identifying as 'Turkish' in the 20th century and their culture displaying a marked influence of Turkish. Even the term 'Turkish' as used by them amounted to nothing more than 'Muslim'. Hence, no 'converted' population or 'Turkish enclaves' in the old Ottoman Empire. The extent of 'Ottoman colonialism' is still a subject of debate among historians, but I think everyone would agree that Turkification is a much more limited and later phenomenon when compared to Europeanisation that came with European colonialism.

That said, the influence of Turkish culture in the Middle East is certainly not negligible. As u/Regalecus and u/ohyeahyeah727 have already pointed out, Turkish cuisine has had an influence on Middle Eastern cuisine at large. The Ottoman legal system remained/remains effective in many of these countries even after it became null in Turkey, which we most readily identify as the direct descendant of the Ottoman Empire. If I am not wrong, certain words like efendim (means what? or sir! depending on the context) especially pertinent to military have made their way into Arabic. The first president of Israel, David Ben-Gurion studied law in Istanbul in 1912-14! Google 'David Ben-Gurion fez' and you are in for a treat: the very first president of Israel wearing a fez in Istanbul. Yitzhak Ben-Zvi, the longest-serving president of Israel studied with Ben-Gurion in Istanbul and also has a fez picture. So the connections and influences are not that weak.

Again, however, you would be right to observe that the Turkish culture has never had the impact on the Middle Eastern peoples that the French had on Africa, say. I think this is, again, in part because Ottomans did not care about having such an influence for a very long time. The French had a 'civilising mission', as they called it, hence they opened schools, deliberately toyed with the idea of creating a Europeanised culture in their colonies, etc. When the Ottoman state officials decided to do it by imposing Turkish education in state schools, for instance, there were already active Arab nationalist movements. Plus, from what I know, Ottoman officials did not really like engaging with local folks most of the time, even in Turkish majority regions.

It is very late where I live, so I hope this comment does not come across as too much of a rambling. Let me reiterate my points. First, the existing Turkish enclaves in the Balkans (and also in Cyprus) were a result of the Ottoman population engineering, which did not apply to the Middle East in any significant degree. Second, the state simply lacked the motivation to Turkify its population for the vast majority of its long history; therefore no Turkish-speaking enclaves that are not ethnically non-Turkish either. Despite all this, however, there are some Turkish influences in the region, as noted by other contributors too.

I don't know about others, but in Israel there are many refernces to Turkish culture, actually. Several Turkish words have officially entered the Hebrew language (Tabu, Dugri, Dunam, Karahana) and are used frequently, some architecture is inspired by Ottomans, Turkish food has become loved Israeli food (Burekas, Shishlik and Baklawa), the Israeli judicial system is based on both British and Ottoman laws... I mean, half of the newer parts of old-city of Jerusalem were built by Suliman the Magnificent, which is a well known figure in Israeli history.

It could be less prominent because the Turks who ruled here used Arabic as the main language, and not Turkish, and were quite Arabized in culture.