Society, for the most part trusts science. As society becomes more educated about the possibilities for growth and advancement, there is more acceptance of science ideas. There are some members of society who are reluctant to trust science. African Americans have been distrustful of science because of events like the Tuskegee syphilis experiments. Religious groups have been distrustful of science because science negates some religious ideas. As we have more and more natural disasters even the most reluctant are beginning to trust more science.
Yes, I believe that society does trust science. Skeptics of scientific observation used to include the majority of society, but as time and science has progressed, more and more people started to realize that things like gravity, nature, and reproducing life forms could not be explained by anything but scientific data and proof. While it's very likely that we still haven't even began to scratch the surface on understanding all the phenomena we see occurring around us everyday, what science has discovered, society agrees with. There are still skeptics, but these are mostly just religious skeptics because if science is indeed correct, it disproves the existence of God and religion is therefore disproved.
Science is the only source of valid facts that exist, everything else is pseudoscience. If we learn not to trust science, we will become ignorant to the true facts around us, and believe in unproved ideas, we will fall victim to the selfish nature of con-artists who will trick us into spending our time and money on fake scientific products.
People aren't science literate enough to know what matters and what doesn't. And they don't want to be - because that's boring and hard and elitist. It's only went they want to prove a point that people turn to science, but because they are illiterate as to methods and meaning they end up blindly holding up junk as gold and the world suffers. So yeah we are in a dark ages where witch hunts are rampant and educational standards are always the first casualties.
Society does not trust science as much as it should, as evidence by the repeated leaps of progress in various scientific endeavors, and lack of social progress. If the work of scientists dictated our behavior instead of appeals to superstition by politicians or religious figures, we might actually see the "change" so many folks like to talk about in our lifetimes.