How is public opinion related to public policy?

In the United States, public opinion plays a significant role in US foreign policies. As an example of this was the war against Saddam Hussein's Dictatorship in Iraq.

Both major allies in this war, the USA & the UK were convinced that Iraq was a threat to world peace.

Public opinion at the time favored military action against Iraq.

As an example of another case we have the US participation in the two world wars.

The USA entered both world wars late after the wars had begun. In both cases, public opinion was reluctant about entering these wars. Finally as we know the USA had no choice to enter the wars as public opinion began to see the need to do so.