I think it is laughable, when viewed against the net of human history, to say that there is a problem with science. The world is increasingly wealthy overall. However, there is a problem in complexity. There is a misunderstanding even among scientists about the fundamental mathematical underpinnings of information. The butterfly effect and the P=NP problem essentially say that, as far as math goes, we don't know what initial dependency might have some severe effect downstream, and that, if there are too many variables, we can't do much anyway.
Yet, politicians of certain political stripes and some scientists themselves are enamored of the idea that we should have "science based" policy making. Policy making is about masses of people, and too many variables. Thus, even though science can say, "these people are less meat based upon and were be better off", science cannot say "everyone will be better off if we eat less meat so let's make it a law". Indeed, there's a baked in butterfly effect that says any public policy has winners and losers. When we make laws that say, 90% of the people will be better off, well, those 10% are going to be irritated. At some point, as a civilization wanders through its history, it accumulates more and more of those people that were screwed by the law. People being what they are, they don't care about how they might have benefited through being in the 90% groups, but how they were in the 10%. If new science proves that the people in the 10% were actually -right-, then, it only makes matters worse.
From a government perspective, we've actually picked the worst things to apply science to. In most people's lives, it is their diet that matters most and the science underpinning FDA recommendations and recommendations from other food authorities has been fabulously and publicly wrong. Many Americans have grown up hearing that first, butter was bad, then, butter was good, then, corn syrup was better than sugar, then sugar is better. First, its clogging of the arteries caused by cholesterol caused by diet, then, just as every middle aged american devours statins, we find out it is a combination of stress and lifestyle. It doesn't help that the public lumps doctors in with scientists - to them, scientists just means "smart people", and they see doctors screw up enough that every family has the story of the loved one that doctors wronged.
The mistrust of the medical establishment when it comes to diet is epidemic and bipartisan. There's plenty of both tree hugging liberals and gun toting conservatives reading about various health food supplement and other weird nonsense about diet and health and even medicine on the internet. The FDA and the food industry alike are seen as corrupt in the minds of both conservatives and liberals is telling. Granted, they filter that corruption into their own political worldview, but that they don't trust these institutions at all suggests a real problem.
From there, it is easy to see, that if the public doesn't believe any of the science about the thing most common in its life, and the institutions designed to protect that science, then, it is going to be a hard sell for the public to genuinely trust science in anything beyond the latest breakthrough to make their consumer products better.