I do not think most people are going to change their minds about this. We have seen extreme cold temperatures so now we know that this can no longer be called global warming. What it is is severe climate change, and that is becoming more and not less evident to most thinking people.
We can restore public trust in the reality of climate change as long as we make an effort to increase scientific literacy in schools. We have allowed science to be devalued by people who are too ignorant to know the facts. This strain of anti-intellectualism in American culture must be vigorously opposed. Also, the media has to stop treating climate change deniers as if their viewpoint is equally valid instead of exposing its demonstrable falsehood.
I believe it is possible to restore public trust in climate change, but only if we shed the prior label of "global warming" as it implies the earth only gets hotter. The cause would benefit from focusing more on what people can do to reduce their impact on the environment, and less on fear-mongering.
There is not enough proof to restore people's faith in climate change because we can only track small climate changes a few hundreds of years back. How do we know if this is just part of the natural cycle or that we are contributing towards it enough to make a difference with no hard evidence of what "normal" is?
No, it is not possible to restore public trust in climate change, because nobody believes Al Gore anymore. If Al Gore really believed in global warming, he would not live in a giant house and travel all over the world. People do not believe that those who believe in climate change are genuine anymore.