The Khazarian hypothesis has been conclusively disproven through genetics. Ashkenazi Jews, whom the hypothesis is about, have been proven to be of Levantine ancestry with some amount (no more than half by even the most generous estimates) of European ancestry.

Genetics tell us that while the male lines remained predominantly Levantine/near eastern, mitochondrial DNA shows more admixture with the European populations (particularly Southern European), indicating that it was more common for Jewish men to take gentile women as wives than vice versa, although they would have had to convert per religious custom. Of course, with the founder effect this could all have been in low numbers.

Nearly all Jewish ethnicities (Ashkenazim, Sephardim, Mizrahim, and others) cluster together as being of common Levantine/Middle Eastern origin, but also cluster with the local populations that surround them or whom they dwelt alongside historically. For example the Jewish communities in the Caucasus are more closely related to the peoples of Iran than to Caucasian ethnic groups, which tracks because their communities immigrated there from Persia long ago. The exceptions are those whom we know have a unique history, such as the Subbotniks who are descended from 18th century converts.

What the genetic and historical evidence tells us about the story of the Ashkenazim is that their community originated with Jews who emigrated from the Levant prior to 1st century AD, fled from what was then Roman-occupied Judaea during the wars of the early 1st and 2nd centuries, were taken as slaves by the Romans during this period, and who emigrated at later points during the Roman Empire. Central and Eastern Europe were far less developed during antiquity than the Roman lands were, and did not have political structures needed to be desirable for immigration. So the earliest European Jewish communities formed in Imperial lands such as Greece, Italy, Spain, and France.

As the Medieval period kicked off, additional parts of Europe became more hospitable places to settle, and for a time Jews were encouraged to settle in the Frankish lands during the Carolingian period, with the Rhine valley being particularly popular. Many were already in France and so there was a great eastward expansion. Eventually, they would expand into areas such as Poland, the Baltic coast, the Ukraine, and Russia, as those areas developed, especially when antisemitism forced them from their homes.

The Jewish communities of Central and Eastern Europe were generally some of the safest and most stable within the Medieval Christian world (though that’s not saying much). Jews from other parts of Europe, North Africa, and the Middle East would often join their co-religionists in those communities, such as refugees from the English expulsion of Jews under Edward I.

The Khazarian hypothesis originated in the 1800’s from no evidence, contradicted known history, and has always been pseudo-history. It is worth noting that today it is mainly cited for antisemitic purposes, so as to separate the world’s largest Jewish ethnic group from their own identity for malicious political or social agendas. I am not accusing you of that, btw, I understand that you’re just asking. But it is worth knowing.

Edit: cleaned up a few sentences