It is far too common that cultural and religious convictions challenge the validity of reliable scientific evidence.
Persecutions, wars, bubonic plague pandemic, managed to drastically diminish all European Populations, including Ashkenazim Jews.
There should be no dispute on the bottleneck genetics Ashkenazi Jewish populations went through around the year 1000 CE.
Further genetic tests showed that two fifths of Ashkenazi Jews are descended from only four women! 
Later rapid population expansion of Ashkenazi Jews, that nowadays constitute the vast majority of all worldwide Jewish populations, does not preclude the fact that they risked extinction during that era.
It is also wrong to claim Y chromosome studies to prove the homogeneity and inbreeding of all Jewish populations, since it is known that it was common for Ashkenazi Jews to convert and marry local young women. Women do not bear the Y chromosome. 
It is also wrong to claim "the presence of the Eastern Mediterranean pool of genes among practically all present day Jews", the belief that all contemporary Jews descend from the ancient Israelites.
Khazars ruled a vast Empire, and were mass converted to Judaism.
Apart from the Jewish religion, they completely adopted the Jewish letters, customs, judicial laws, education, etc.
At the times of the final invasion, submission, and decline of their Empire, Khazars numbered many millions, and were all converted to Judaism. No genocide was reported, they were just displaced all over Europe. They did not vanish, most Yiddish words and the grammar Ashkenazi Jews use has ancient Khazar roots. 
Competing interests: No competing interests