Writing with an American perspective:
The graph of decline in trust in the media over time is interesting. I'm not sure it's necessarily because the media is less trustworthy, although that's probably the case given increased partisanship about every subject nowadays. But like "yellow journalism" is a term from the 19th century; this isn't a new phenomenon. I think a better explanation for the decrease in trust in the media is an increase in information enabled by the internet. Sometimes this information is false and the media is reporting the truth, but people are more willing to believe what lines up with their narrative so seeing the media disagree with their belief decreases their trust in the media. Sometimes information from outside the media is true and the media is reporting falsehoods; this also damages people's trust in the media's credibility. This is a feedback loop then where instances of false reporting cause doubts on instances of thurthful reporting, and instances of truthful reporting that don't align with a group's biases are attacked as possibly false.
Back when there was only one real source of truth, the media, any alternative sources of information were so few in number that their combined efforts couldn't induce any doubt in the media's veracity. At that point the feedback loop couldn't get started, so it didn't matter if what the media reported turned out to be true or not. We romanticize that time period by assuming that journalists of the time were more trustworthy; that may or may not be true, but I have a hard time believing that human proclivities -like twisting the story to fit the journalist's biases or to hide stories that go against the narrative- have changed significantly for the worse since then. Journalists of the time probably did many of the same things as modern media is accused of doing, but without anyone else to point it out there was no decrease in trust.
The media has mostly responded by trying to insist that they're always trustworthy. This probably isn't the best move. It definitely increases partisanship as it moved people into one of two camps: believers of everything the media says or disbelievers of anything the media says. I don't know what a new agency that somehow emphasizes that it's not totally trustworthy or that its articles may be flawed looks like, but it'd probably get too much flak from both camps to survive currently. So I think that "fixing" the media involves a decrease in partisanship first.
(In my opinion the biggest thing we could do to decrease partisanship would be to revamp the political system to favor smaller parties that have to compromise and form a coalition to govern. The current US voting system, first past the post, mathematically favors two parties only (https://ncase.me/ballot/), so the first step would be to reform the voting system to something that allows third and fourth and fifth parties.)