Firstly, healthcare is not a free market like, say, the market in recorded music. People don't wake up one morning unable to function because they haven't got the latest album by their favourite artist. They are free to or not to enter the recorded music market at no cost. Contrast this to the market in healthcare where very often people have to choose between suffering and/or dying, or entering the healthcare market. There are differences on the supply side as well. I can set up a shop selling recorded music tomorrow if I want, but the supply of medical professionals is severely limited by the need to train and qualify from a medical school and obtain a medical license. I am prevented by the law from offering medical treatment to the public. These two factors ensure there will NEVER be a realistic chance of free market in healthcare emerging, no matter how much you want there to be one.
Secondly, people respond to incentives. Pay medical professionals per unit of care and they will strive to produce lots of units of care until producing an extra unit costs more than doing something else (e.g., playing golf or sailing one's yacht). Similarly, sick people will put off seeking medical care until their condition deteriorates to the point that the costs of not being treated are equal to the costs of being treated. It's a well established fact that virtually all medical conditions are less expensive to treat the earlier they are detected. So all co-pay does in reality is provide medical professionals with patients who are sicker when they come to see them and so start supplying more expensive (and profitable) units of medical care from the outset. There would be a temporary decline in healthcare usage while the level of sickness increases to the threshold and then the same or greater demand as before would ensue.