The 'Jews control the media' conspiracy is basically a subset of the secret society theory, which has been around for hundreds of years in various forms, and has often involved Jewish control. Jesse Walker called it the "Enemy Above" class of conspiracy, and Barkun the "Systemic Conspiracy" theories.

The short answer to how it survived the reveal of the holocaust is, anti-semetic groups and individuals denied and continue to deny that the holocaust occurred at all. Especially after the founding of Israel, many displaced people and several scholars in both the middle east and western world like Bradly Smith began to express doubt that the holocaust occurred at all, and if it did that it must have been exaggerated. Once you believe that, it seems almost logical that if the Jewish people were able to pull off such a massive fraud in order to retake the Holy Land, they must have strict control over the media. Among holocaust deniers, those who publish openly questioning materials are seen as courageous, and anyone in the media who attacks their arguments is in on the conspiracy or so brainwashed that it amounts to the same thing. Since the holocaust, the theory has spread further than just the holocaust denier groups to become a relatively mainstream conspiracy theory again. There are also conspiracies that 9/11 was in truth committed by Jewish groups, usually Israel, and that this was suppressed by the Jewish control of the media.

It's hard to get accurate information as to why conspiracy theories spread and persist, there is still debate among anthropologists as to what purpose if any they serve in local and global politics and what effects or meaning they have in terms of the psychology of the individual who believes them. There are of course other factors in play than I have talked about here, but anti-semitism did not die out, even temporarily, after the holocaust.

As a follow up, how did it begin before WWII?