If the Khazar Hypothesis is true, we should see Central Asian genetic material in Ashkenazi Jews on the order of 25-50%; and among their Aaronic priestly class, we should see the Cohen Modal Haplotype at no higher than the base rate of 5-15% for the broad region of the Mediterranean, Arabian, and Caucasus regions.
If the Rhineland Hypothesis is true, we should see very little Central Asian genetic material in Ashkenazi Jews and there should be at least some elevation in the frequency of the Cohen Modal Haplotype, maybe even a lot.
What we actually see, now that we can measure it, is that the amount of Central Asian genetic material among Ashkenazis approaches zero, and the Aaronic priestly class is 50-70% Cohen Modal Haplotype.
The Khazar Hypothesis is therefore not true, and it's not close.
The Rhineland Hypothesis might still fall to some other explanation, but Khazar ain't it.
Enough of my audience has some contact with fundamentalists that they may encounter folks who believe the Khazar Hypothesis, that is, the belief that Ashkenazi Jews from Eastern and Central Europe are not genetically descended from the Middle Eastern Jews of the Bible, but from a North Caucasus group called the Khazars, a Turkic people who came out of Central Asia and conquered many peoples in the regions of what are now Georgia, Ukraine, Kazakhstan, and Russia up as far as Volgograd, forming a Khaganate in the 7th-9th C AD. They converted to Judaism, or at least the ruling elite did.
With the fundamentalists of this sort that I have known or have read, the conversation can quickly go to assertions that the Jews of Israel (and New York) are therefore not the ”real” Jews, not the Chosen People, and we are therefore free to believe that they are deeply involved in the conspiracies of world domination and no friend to Christians. Old-line Christian fundamentalism did not have the pro-Israel stance that Evangelicals have today, but had deep suspicion or even hatred of Jews. This was strong among southern Baptists, and can be seen in the early comments from Jimmy Carter, talking about how Israel only prospers when it has turned itself over to God and should not be attempting to defend itself outside that context. (As a side note, I think that anti-Semitism is different in different parts of the country. I should probably think about that and write it up.)
I am only giving an overview here, because I am no expert. I am only able to give you such information as will reassure you that this is nonsense. If you get into a protracted discussion with such people, you will need to be better armed with more detailed information. I can put you in touch with that information quickly, BTW. With graphs and pictures!
It is fair to note that while most people who believe that those Jewish converts are the actual ancestors of modern Western Jews are anti-Semites and cranks, not all of them are. The idea has a long, complicated history that has been embraced by some Jews and those sympathetic to their protection, and a few of the strongest voices in favor of the idea are legitimate scientists. Now that the discussion has moved to genetic/genomic evidence rather than the hodge-podge of historical, linguistic, and speculative sources that kept the idea alive for centuries, however, the main researcher with any credentials and credibility who believes in the Khazar Hypothesis is Dr. Eran Elhaik of Johns Hopkins and those associated with him. Even he has modified his view to a more Turkic-Anatolian Hypothesis, though still related to the Khazars. In history, the Khazar Hypothesis has attracted some interesting thinkers. But today, it is Elhaik and the linguist Wexler* versus everyone else in science.
The prevailing theory is the Rhineland Hypothesis, that Jewish traders in the Western Mediterranean intermarried with Southern European women, perhaps as specifically as Lombards in Northern Italy, and moved to the Rhineland in the 9-10th C to expand trading there. There may have been some who came even earlier. An initial population of about 500 eventually expanded to the millions in Eastern and Central Europe in the 20th C, concentrating at first in Lithuania and Poland. They did not tend to intermarry with any of the local populations; those who did left Judaism and joined the surrounding culture. The Jews who later came to Western Europe and the Americas were drawn almost entirely from this group.
We now have the numbers, and subject to some variation, modern Ashkenazi are about 55% Middle Eastern, 35% Southern European, 10% Northern European, especially Slavic. That is just about exactly what we would expect to support the Rhineland Hypothesis. It gets more interesting. The Y-chromosome haplotype groups are strongly Middle Eastern, while the female uniparental lines have strong elements of Southern European mtDNA. Those also have lines common to Middle Eastern and Slavic lines. This is not a picture of some Jewish guys who set out from the Levant, pick up some wives in Lombardy and head across the Alps. Jewish traders had been in the Western Mediterranean for centuries, and might not have thought of themselves as “belonging to” a physical Israel that no longer existed. But their DNA would still have been from Palestine.
It gets stranger still. The priestly line of Cohens (Kahn, Cohn, Cohane) is inherited father to son, as is the Levitical line. (Levy, Levine). These lines are preserved not only through the last thousand years, but 2-3 times that long, as the priestly classes in Jews in Persia, Yemen, and Southern Africa have a strong concentration of the same Y-chromosomes - an R1a1 variant for the Levites, and a J1 variant that is even called the Cohen Modal Haplotype. Among the people who call themselves Cohens or Levites today, 50-70% have those lines. Very few paternal irregularities there, even over thousands of years. Pretty impressive. They did protect the role tightly. The R1a1 is relatively common over a wide area, but the J1-538 is not. No Khazars.
There are other lines common among the Ashkenazi, such as E1b1b1, which are spread among many peoples. You might find some Khazars there, but there’s no specific indication of it. When they founded their kingdom in the North Caucasus they ruled over, and probably absorbed, Georgians, Armenians, Circassians, and other groups. Not all R1a1 lines are Levites, either, not by a long shot, and those lines today could come from any of a number of places in the region. Also, the Khazars must have gone somewhere, so why not Eastern Europe and Russia? It’s an easier trip in many ways. That has always been plausible, even though there is no record of it. But now we have the data, and they do not seem to have done this.
Genetics and genomics are better for disproving hypotheses than for proving them.
Remembering our history that the Sephardic Jews came from those who were expelled from Spain and Portugal 500 years later, who mostly went to Morocco and the rest of Northern Africa, we would expect them to be genetically similar to Jews from the Western Mediterranean who married local women. In fact, they are. The Ashkenazi and Sephardi are more strongly related to the Druze and Palestinian lineages of today, less strongly to Iranians, Iraqis, Syrians, and not at all to Central Asia. No one has ever suggested the Sephardis are connected to the Khazars.
A note on Eran Elhaik’s choices in his research, which most others think is where he started to go wrong. We can test the DNA of modern groups, but not historical ones. Jews were settled all over the Roman Empire (and more), and those numbers increased after the fall of the Second Temple, but we don’t have their DNA. Nor do we know what the DNA of the Khazars was. In both cases we have to use proxies, of populations now living who show some continuity back to those areas. In the case of the Khazars, Dr Elhaik chose Armenians and Azerbaijanis as his proxies, largely because they are from that area. As he also included the Jews from those areas as part of his typicals, that would artificially tilt the sample toward the Palestinian Jewish lines, but calling them Khazars to buttress the similarity with modern Ashkenazis. Yet we have already seen that even though the Khaganate was in the North Caucasus, the Khazars themselves were from Central Asia. Not a good match. To try and match for Palestinian Jews of 2000 years ago he chose Bedouins and Hashemites, which are not terrible choices, but leave out others such as the Druze.
*Wexler believes that Yiddish has a foundation that is closer to some Caucasian languages than to Germanic ones, though the German later became dominant. He bases this on some verb structures. It is beyond my knowledge, but I read that those differences also exist in Slavic, which is already known to have influenced Yiddish. So Wexler's idea remains possible - it is not disproven - but it is also not compelling.