No the Khazars are related to the Turks, and have slightly(By slightly I mean insignificantly) altered the Ashkenazi Gene Pool.
But Israeli historian Schlomo Sand says,
"The most significant mass conversion occurred in the 8th century, in the massive Khazar kingdom between the Black and Caspian seas. The expansion of Judaism from the Caucasus into modern Ukraine created a multiplicity of communities, many of which retreated from the 13th century Mongol invasions into eastern Europe. There, with Jews from the Slavic lands to the south and from what is now modern Germany, they formed the basis of Yiddish culture."
"Since the 1970s supposedly scientific research, carried out in Israel, has desperately striven to demonstrate that Jews throughout the world are closely genetically related."
"By validating an essentialist, 
ethnocentric definition of Judaism it encourages a segregation that separates Jews from non-Jews – whether Arabs, Russian immigrants or foreign workers."
"The Israeli forces who seized Jerusalem in 1967 believed themselves to be the direct descendents of the mythic kingdom of David rather than – God forbid – of Berber warriors or Khazar horsemen."
sourceThis historian apparently disagrees with your "take" on Jewish history, Oxy. You might want to check him out.