There are three ideal or pure kinds of economic systems. These are traditional economies, command economies, and market economies. In a traditional economy, tradition completely dictates what is made, how it is made, and for whom it is made. In a command economy, the government controls those decisions. In a market economy, only the market determines those things. In the US, the market does not decide each and every one of those things and, therefore, we are not a pure market system.
Think of all the ways in which the government decides what can be made, how it will be made, and for whom it will be made. The government determines that we will make tanks for the military. It determines that we will not produce cocaine. It does not simply let the market decide what will be produced. The government says that all cars will be made with seat belts. It says that all things made in the US will be made by people making at least minimum wage. It uses programs like Medicare to decide who will get certain goods or services.
Therefore, we cannot say that we are a pure market economy. In a pure market economy, the government would never intervene to make the sorts of rules or create the sorts of programs mentioned above. That is why it is proper to say that we have a mixed economy.