I'd like to highlight this previous answer by /u/gingerkid1234, which discusses in details the many problems with the strong 'Khazar hypothesis' and why it isn't entertained by historians. Gingerkid1234 additionally has [this answer] where he discusses how the Khazars don't serve as a reliable source for the bulk of eastern European communities (let alone the rest of Europe), but that some Khazars may have been integrated with existing Jewish communities after their conversion.

Additionally I'd like to mention /u/yodatsracist, who responded to the first linked post talking about how hard it is to get primary source material relating to the Khazars, and has written before on matters of religion and ethnicity.

This assertion that Ashkenazim are 'actually Khazars' seems remarkably resilient, despite the complete lack of evidence for it (antisemitism certainly plays a major role.) I would appreciate it if someone could discuss why this issue persists in discourse, but I'm not sure if that would qualify as requiring a new post.

https://www.reddit.com/r/AskHistorians/comments/92owbf/did_the_khazars_really_convert_to_judaism/

/u/gingerkid24 has a great answer dealing with the uncertainty about the khazar's religion and refutes the conspiracy theory directly

https://www.reddit.com/r/AskHistorians/comments/399uoi/when_did_jews_migrate_to_eastern_and_western/

You might find this comment by /u/idjet interesting too