Was the Turkic Khazarian empire the only non-Semitic Jewish nation?

Khazaria was NOT a Jewish nation! Given that fact, Turkic Khazaria couldn't have been a non-Semitic Jewish nation.

The idea first got widespread attention from Arthur Koestler's book, "The Thirteenth Tribe" in 1970-something, it has persisted. 

Many current proponents find such an explanation for Jewish ancestry to be a politically expedient justification for redrawing territorial boundaries in the Levant. Others, including some Jews, find the Khazar-origins idea to be interesting and compelling because, well, it is!

The Jews-as-Khazars idea has been thoroughly debunked by modern genetics:

Plausible at the time, the Khazar-origin premise has crumbled under the onslaught of modern molecular genetics. The latest volley: a study published this week in Nature Communications. The study’s senior author...[uses] high-resolution techniques that have highlighted the historical hopscotch of other migratory peoples... The authors analyzed the Y chromosome – a piece of the human genome invariably handed down father-to-son – of a set of Ashkenazi men ...

If Ashkenazi Jews were the progeny of Khazar royals, their DNA would show it.

--New genetic study: More evidence for modern Ashkenazi Jews’ ancient Hebrew patrimony via Stanford University School of Medicine

The same Y chromosomal signature was found in every one of a large sample of Ashkenazi Jews, and quite a few non-Ashkenazi Jews. However, it was not present in the Y chromosomes of modern European non-Jewish men, nor in male inhabitants of what was once Khazaria.