I think there is far-right-wing propaganda like you're describing, but that doesn't explain why moderate liberals feel like the media is strongly biased in favor of the progressive left. Of course, many on the progressive left think that moderate liberals are holocaust/climate-change deniers, so this may not be very compelling to them.

> moderate liberals feel like the media is strongly biased in favor of the progressive left

I haven't heard that. Do you know any research on it (edit: I understand if not)?

IME moderate liberals just adopt a position between the left and right, regardless of the positions. It's navigation, not opinion.

The progressive left thinks the NYT, for example, is conservative and corporate.