The Islamic propaganda is focusing on the 1950's book by Arthur Koestler, a Hungarian Jewish writer, "The Thirteenth Tribe". Koestler wrote this book after the war, soon after realizing that all his relatives had been killed by the Nazis. He was desperate and trying to find a magic way, an argument with God, that would turn history back (a fantasy most European Jews of my age share) and to convince God or whoever did it, that the Holocaust was a mistake, that European Jews were not really Jewish, that we were a Turkish tribe from the steppes like the Hungarians and had nothing to do with those hated Semites. The Nazis had not killed the Turkish speaking Karaites (a few thousands) accepting their lfie-saving argument that they were converted Khazars. Of course they were not, but in that situation it was question of life and death for them. As soon as the Nazis were gone and aliyah made possible, the moved to Israel en masse and the Khazar story was forgotten. Not even Ahmedenijad and other anti-Jews remember that "real" Nazi-certified Khazars are living in Israel and are Zionist as the most.
The problem with the Khazars is that they disappeared 500 years ago without leaving a trace. We dont even have a description of them. Were they brown skinned and slit eyed like the Mongols, or were they red haired and blue eyed like the Goths (that lived at that time in the same area - the valley of the river Don)? We have no idea. There are no Khazar ruins, no Khazar books, no Khazar legends. For centuries it was believed they never really existed, all that was a vague and improbable legend of a far-away kingdom that converted to Judaism. All we have is an Arab traveler's description of his visit to the main campsite (capital) of this nomadic tribe, but it is very obscure and deals more with the Arab himself than with his boring hosts. The nearest probable surviving relatives of the Khazars are the Chuvash in Siberia. They are a few thousand trappers and hunters in the vast Russian forests, not known for their intellectual achievements. How could the German-speaking study-loving uber-intellectual Lombard-looking Ashkenazi people be their closest relatives?