The United States is not a pure democracy, it is a republic. Accurately defined, a democracy is a form of government in which the people decide policy matters directly. A republic, is a system in which the people choose representatives who, in turn, make policy decisions on their behalf. The Framers of the Constitution were altogether fearful of pure democracy.
Read the federalist papers, they believed pure democracies have in general been as short in their lives as they have been violent in their deaths.
By popular usage, the word democracy has come to mean a form of government in which the government derives its power from the people. In this sense the United States might, though inaccurately be called a democracy.
Some exceptions would be where states allow voters to directly vote on propositions, they are example of partial true democracy.