The Jewish diaspora has had many waves since pre-classical antiquity, starting with the Assyrian (~722 BCE) and Babylonian (~586 BCE) exiles. Since then, Jewish people gradually spread throughout the Hellenic and Roman worlds. The earliest distinct Ashkenazi populations coalesced in the Rhine valley around the turn of the second millennium from Jews migrating northwards through the lands of the Holy Roman Empire (which under Charlemagne and his successors granted Jews more social, religious and economic freedoms than most other medieval polities), and since then their history has been pretty well documented.

There is no evidence that Ashkenazim have any significant genetic connection to any Turkic populations, nor is there any archaeological evidence that would substantiate a widespread conversion of an entire kingdom in the Caucasus region in the late first millennium. There are several textual sources that point towards some kind of Jewish leadership or claim the existence of a Jewish polity in the Near East — a couple of letters and a couple dozen second- or third-hand reports — but there's no more historical context for these than there is for many other medieval sources.

While it may have started as a good-faith (albeit distinctly light on historical rigor) effort to identify Ashkenazi ethnic origins, the Khazar myth has been little more than an anti-Zionist conspiracy theory since at least the late 19th century. Even if there was a Jewish state in the Caucasus region during the Middle Ages, the above-cited genetic study and others suggest that any purported connection between the fall of the Kingdom of the Khazars and the rise of the Ashkenazim is just a coincidence. Correlation does not imply causation.

I'll add, at the risk of getting too polemical, that imbuing historical coincidences with such deep, sweeping significance is one of the hallmarks of conspiratorial thinking, and it has far more to do with the conspiracy theorist's own need for ideological validation than it does with anything even remotely resembling historical rigor or objectivity.