Such events as the conversion of an entire population to another religion don't happen at once. I will try to break this up into several parts.

Background

During the Mazdakite movement in Sassanid Iran, the Jews supported Mazdak and fought against traditional Zoroastrian Persians. After Mazdak had been killed (~530 AD), Jews in Iran were persecuted and many had to flee to the North. So Jews indeed had been living in the regions that later became Khazaria since the VI century. Many had become influential merchants and had certain weight in the Khazar society.

On the other hand, spread of Christianity and Islam as the new major religions posed a challenge to the Khazar rulers; converting to one of these religions would put them under the influence of other countries. Adotping Judaism allowed the Khazars to be recognized as non-pagans, at the same time remaining independent from major Christian and Muslim powers.

Conversion

There are three different sources describing conversion of the Khazars to Judaism:

1. Letter of Joseph ben Aaron, one of the last Khazar rulers to Hasdai ibn Shaprut, a Jewish scholar from Cordoba Caliphate. According to it, Bulan was the first Khazar king (kagan) who converted to Judaism. Both Christians and Muslims had been disturbed by such news and sent ambassadors who tried to convert Bulan to their faith. During disputes, Bulan posed questions in such a way that the Christians agreed that Judaism is better than Islam and the Muslims agreed that Judaism is better than Christianity. Thus he exalted the Jewish religion as the best one, even according to Christians and Muslims. Then he was circumcised.

2. Letter of an unknown Khazarian Jew (Schechter Letter), also referred as Cambridge anonymous. According to him, Jews fled to Khazaria and assimilated within the local people. They forgot their traditions, except for circumcision and Shabbat. One of such Khazarian Jews became a military leader and urged people to return to the roots and live according to the Jewish tradition. Greeks and Arabs had sent their ambassadors, trying to prevent the spread of Judaism but didn't succeed and all Khazars converted to Judaism.

3. According to al-Bakri, Arabic writer from Cordoba, Bulan first converted to Christianity, then realized he made a mistake, organized a religious dispute which resulted in his conversion to Judaism.

Therefore, the religion of Khazar rulers and elites is not questioned at all, the subject for discussion is rather the exact story of conversion and whether they had any Jewish ethnic roots.

What about the commoners?

No archeological evidence has been found to claim widespread adoption of Judaism in Khazar Khanate. Most Khazars kept their traditional pagan beliefs. There are Christian and Turkic Pagan burials, attributed to Khazaria. Therefore, the current prevailing opinion is that only Khazar elites were Jewish, but not the majority of people.

Civil war

We have very scarce and inconsistent information about the civil war in Khazaria in the IX century. Most of this was reported by Byzantine emperor Constantine VII. The war broke out after the reforms of Obadiah, Bulan's successor that strengthened the position of Judaism and started oppressing Christian, Muslim and Pagan Khazars. Though we know very little about the subsequent events, a lot of people fled or died and the war ended up with victory of the central government that established a strong mercenary army, becoming completely independent from the common people. After that, even though Christianity was frowned upon by the Khazar rulers as a potential factor of Byzantine influence, conversion of all people to Judaism was never enforced.

 

Sources:

Michail Artamonov - History of the Khazars

Valentina Flyorova - Judaism in Steppe end Forest Steppe Khazaria

i wonder