This isn't a problem of democracy. It is far more general than that: it's a problem with humans. It persists even on the small scale -- in an oligarchy, and even in a monarchy or a dictatorship. Indeed, it might be an even worse in a dictatorship, because when the dictator decides to prefer an intuitive falsehood to a counterintuitive truth, there's no effective way to argue.

The answer to this problem, at every scale, is marketing. That's not what a roomful of rationalists wants to hear, of course, but it is the correct answer: Although it is wise to support proposals that are backed up by statistics (because those are the ones that are likely to work!) you also need a sales pitch. You must get people to associate the proposals with mom and apple pie, or with sexy people, or with Progress. Or you must convince them to follow the herd.