> the amount of warming has been consistently underestimated by the IPCC reportsI think you mean overestimated. The IPCC reports have consistently predicted more warming than has actually occurred; by now the actual warming is outside the lower end of the IPCC's confidence interval. The IPCC AR5 responded to this by no longer claiming that its predictions of warming were based on climate models; instead it said they were based on, in effect, the personal opinions of the report writers. This does not strike me as a valid scientific method.
> The press, being the press, tends to sensationalize science with "Conflict! Intrigue! Breakthrough! Physicists vs Biologists! SUNDAY! SUNDAY! SUNDAY!". I suspect that these misleading characterizations have a bigger impact on people's perception of Science than the actions of scientists themselves.
While I agree that the press likes to blow everything out of proportion, I think scientists have to bear the blame as well. Scientists do not do a good job of conveying the relative levels of confidence in different claims; to the public, they portray every claim as being made by Science, with the same authority, regardless of how solid it actually is. It is hard to blame lay people for losing trust in science when so many of these claims turn out to be wrong. What scientists should have done is to not have made those claims with the authority of Science in the first place.
The recent "replication crisis" is an example of this. The reason all those findings weren't replicated is that they weren't valid in the first place; they were only indications that needed further investigation. The scientists publishing them should have known that and should have said so. The narrative that should have been given to the public is "these are interesting indications but that's all, they might not pan out, more research is needed...well, we did more research and it didn't pan out.". But the narrative that was actually given to the public was "WoW! Great new finding! Everybody change everything you're doing!...oh, wait, turns out it wasn't true after all." After enough iterations of this, naturally nobody trusts what scientists say any more.