Shows like Last Week Tonight are the liberal alternative to the right wing nutjob media sites like Breitbart and infowars, which are clearly propaganda and fake news sites.
Thing is that liberals like a lot of fact based media coverage and John Oliver brings that, you may not like his interpretation of events but they are fact based.
I would say that shows like Last Week Tonight and Full Frontal with Samantha Bee are more trusted that regular news sources, at least for liberals. Too often mainstream news covers stuff that is clearly bullshit, lies and propaganda and don't even bother to call it out for what it is, that doesn't happen with the liberal infotainment shows, in fact they base their shows on calling those out.