The problem is that in the 19th and early 20th centuries, Turkey and the Balkans were gripped by a nationalist fever which led to an upwelling of pseudo-history and other issues both in actual educational system and scholarship in that region that causes the issues with studying many of their peoples' ancestry today. For example, Anti-Turkish sentiment today in Bulgaria is a major root cause for the crazy and incorrect etymology and misinterpretation of primary sources to try and show that the proto-Bulgars were Indo-European/Indo-Iranian rather than a proto-Turkic-speaking people. Conversely, you have the same issue with Turkic scholars and education, who often teach Pan-Turanism (a theory started under Ataturk which claims everything comes from Turkic peoples, much like Albanian nationalist pseudo-history) or at least results in claims of Turkic ethnicity when those peoples were not.

What you're talking about is the latter case. Loulan was a small kingdom in the Tarim Basin (Xinjiang or "Western Turkestan") that existed from the 2nd-1st centuries BC, when it was vassalized by the Xiongnu and then the Han. It was a rather diverse place, but its base ethnolinguistic identity suggests they were a Tocharian-speaking people, with Gandaharan and Sogdian (Eastern Iranic) communities. Burials from Niya and other sites nearby do show a presence of steppe nomads, probably an Indo-Iranian speaking people known as the Wusun (who were probably related to the Alans), in the region. There have long been claims of a connection between the Wusun and the proto-Turkic peoples (namely the first "ethnic Turks" of the Ashihna and Gok-Turks), but these claims have largely been rejected by modern, mainstream scholars like Peter Golden, who's probably the foremost expert on Turkic peoples in the United States.

First of all, early Turkic peoples spoke a different dialect of Turkic, known as "Oghuric" Turkic, a remnant of which is preserved today among the Chuvash in Russia. This language is evidenced as early as the 4th Century AD, when a people known as the Chi'eh, possibly part of the Xiongnu confederation, sacked Luoyang and Chang'an in 311-316. A surviving fragment of their language exists in the sources discussing the attack, which confirmed several theories about the structure of proto-Turkic and Oghuric Turkic (and raised many new questions). However, at this time, no people identifying as "Turks" existed. While the Huns who entered Europe spoke a dialect of Oghuric Turkic, and were probably a people descended from the Xiongnu (who were probably Yeniseian-speaking, not Turkic speakers), they were not Turks. The Romans, in fact, even explicitly differentiate them from Turks, with Procopius and other authors recognizing a difference between the Hunnic and Gok-Turkic languages (although there may be a new problem with that, which I'll get to in a minute).

The early Turkic peoples, known as the Ashihna, and later the Gok-Turks, first emerge some time in the early 5th century in the Altai mountains according to Chinese sources. It has been suggested that the word "Turk" comes from Chinese "Tujue" as they were noted for their helmet smithing abilities - while the evidence for their name's origins is contentious at best and heavily debated, the Altai mountains were a major helmet manufacturing region, as will be eventually shown in upcoming scholarship by helmet expert Adam Kubik (unfortunately I cannot share more about these new finds and research). The Ashihna eventually overthrew their Rouran lords and conquered the Mongolian Steppes, forming the Gok-Turkic Khaganate, and later expanded as far west as the Caspian Sea, where they came into Roman contact. "Shaz" or "Common" Turkic, which includes Oghuz Turkic, spread with this expansion, which differs from the "Lir" or "Oghuric" Turkic languages by pronunciation and some grammar word structure. However, the issue is that new evidence from Khuis Tolgoi suggests the Gok-Turkic khaganate used a para-Mongolic dialect related to Xianbei and Middle Mongolic, rather than a Turkic dialect, or at least the aristocracy did.

The Oghuz Turkic people first emerge in literary sources in the mid-8th century AD, 800 years after the Loulan kingdom is vassalized by the Han dynasty, and were found in Transoxiana, between the Caspian and Aral seas. The word Oghuz, like Oghur, means "Tribe," and the term was rather generic. They migrated out of the Altai in the 6th-7th centuries, and eventually many converted to Islam. The term "Turkoman," etc. refers to these Islamic converts. The Oghuz people living among settled cultures eventually dropped their old ethnonym and adopted the term "Turkoman/Turkman," and many settled in what is now East Anatolia, Armenia, North Iraq, and Azerbaijan extensively after defeating the Romans and exploiting their internal upheaval after the Battle of Manzikert in 1071. Many Romans would also marry and have children with these incoming Turkic settlers, who became known as Turkopoles (for being half-Roman, half-Turkish). Generally speaking though, most people settled in what is Turkey today are basically the same as the peoples who have always been settled there, who simply have adopted new ethnonyms (Roman, Turk, etc.) over the centuries as a new conquering aristocratic class assimilated into them. Although these migrants brought *some* new genetic and ancestral elements, and new cultural practices, they fundamentally had overall very little impact on the composition of the local population. The identities and ethnicities that transformed or formed out of their movements are much more important than the genetics or lineages anyways.

I hope this clears things up.

Here's some further reading (Although some of this is a bit outdated now):

https://www.academia.edu/12545004/An_Introduction_to_the_History_of_the_Turkic_Peoples