All the political arguments for restricting media because of a purported link between media violence and real violence are based on studies finding a correlation between the two phenomena. But correlation is simply two things happening in proximity, at the same time, in the same person or people. One of those things does not necessarily cause the other. The alarm clock ringing at six a.m. does not cause the sun to come up. In fact, determining when correlation can be read as causation is a crucial and controversial issue in every science.
Correlation is a first step, like a red flag, of a possible relationship that's worth investigating.
After that, many research designs that go beyond correlation are organized. These better studies feature prospective, longitudinal designs and designs that try to avoid various biases that can produce apparent but mistaken causality. Only after many, many studies have been done, by different investigators, using different designs, with many arguments about possible other explanations for the relationship, is causal relationship even plausible. After several studies, the evidence of a correlation between media and violence is still weak. Therefore, a causal relationship isn't plausible.