Well, I don't know what you've been reading, but there are a couple of facts that militate against Judaism being widespread among the Khazars.
First of all, if the entire Khazar nation had converted en masse to Judaism (a historical rarity, even among missionary religions such as Christianity), where did all of the Jews go? Are we to assume that nearly all of them converted to other religions? That's what it would take, if we assume all of the Khazars converted, because there just aren't that many Jews today to support this theory.
Second of all, the one Jewish group that has a somewhat legitimate claim to being descended from the Khazars - the Karaylar - are numerically insignificant and practice a religion which is so different from normative Judaism that they were actually excluded from Russian restrictions against Jews and later the Death Camps. The evidence suggests that the Khazars were Karaites, which brings me back to my first question: where did all these Jews go? There aren't that many Karaites around. Arguing that the Khazars were the source of European and Russian Jewry is a bit like arguing that the Celts were the source of Europe's Protestants, because some Roman Catholics happen to be Celtic, and Roman Catholics and Protestants are both Christian.
Third, if the (numerically more significant) Ashkenazic Jewry of Russia were descended from the Khazars, how did nearly all of them come to speak a dialect of Medieval German, despite the fact that nearly none of them lived under German rule? Wouldn't it make more sense for them to speak some Turkic or Slavic dialect, instead of the utterly alien German language, whose nearest speakers were miles and miles away?
Finally, most of Koestler's primary data on the Khazars (and that of subsequent scholars) comes from Abraham Firkovitch, a lay scholar and historian who traveled the Caucasus collecting information on the Karaites in general and the Khazars in particular. In his quest to prove the Khazar origins of his own sect (the Karaites) he forged many tombstones, inscriptions, and other documents. Thus all of the data about the Khazars is somewhat compromised by Firkovitch's "research."
In any case, I've seen Palestinians who were dead ringers for Jews, and vice versa. In fact, all Levantines have a very "Mediterranean" European look to them. Judging by looks alone is no way of ascertaining someone's genetic makeup.