http://www.khazaria.com/khazar-quotes.html
Koestler's claim that all or most Khazars became Jews is false. In fact it is refuted in the primary sources. But some Khazars did convert and according to Khagan Joseph, they intermarried refugees from the Caliphate and Byzantine Empires.
Nobody disputes any of this. What is in dispute was A. whether conversion to Judaism spread beyond the royals and nobles of Khazaria to the general population and B. regardless of A, what fraction of modern day Jews are actually descended from Khazars.
There isn't much evidence of a Central Asian "genetic signature" among Jews. The male-inherited Y-chromosomes point of Middle Eastern origins (similar to Druze, Lebanese Arabs), the female-inherited mtDNA points to European origins (related to Italians). If most modern Jews were Khazars, you'd expect their haplotypes to cluster with Georgians, Armenians, etc, which they generally don't.
> Koestler’s claim that all or most Khazars became Jews is false.
That’s correct, and that’s what I’m saying — the idea that European Jewry has *no* Middle Eastern roots is Koestler’s real underlying bias.
Conversion to Judaism has been rare, but obviously has happened.
That “khazaria” site is an ethnocentric agitprop site — “Over a thousand years ago, the far east of Europe was ruled by Jewish kings who presided over numerous tribes, including their own tribe: the Turkic Khazars.”