This is a profound question, and its answer is vitally important to choosing the correct path for future media creations. It is unlikely that I could definitively answer the question here, nor am I attempting to do so. The purpose of this post is merely to examine the question.
For the sake of this subject, it is important to point out that there are two classes of media possibly at fault. The first is fictional media—stories of various types largely conjured by the media creator from a patchwork of imagined ideas, personal experiences, true stories, and other fictional stories founded on often unknown origins.
The other is non-fictional media of almost any form, which (if it is truly non-fiction) is based solely upon facts. These facts can range from live footage of an event to interviews with eyewitnesses, and from documentation of evidence to examination of documents or other information media which is relevant. The results of such a basis form a coherent presentation as a documentary film, journalistic story, or truthful dramatization.
In the context of this question, both types of media are generally of the visual kind, such as TV news programs, TV series, films (both entertaining and informative), and video games (typically classified as fiction).
So then, what do I mean by "violence"? Wiktionary defines the word as an: "Action intended to cause destruction, pain, or suffering."
First, I will set aside the argument that (according to this definition) fiction cannot be violent. One could argue that the actions which are portrayed in media are not in reality intended to cause harm (i.e.: to the other actors or stunt people). However, this argument is largely a semantic one, because it is not the action itself which is in question but the image of the action. This image has the same effect regardless of its media classification.
But what exactly is that effect?
There are three general effects violent media could have on society. It could cause an increase in the instances and degree of violence, which is most commonly asserted. It could have no effect whatsoever, being regarded as separate from life entirely. It could also cause a decrease in the instances and degree of violence, which is not a common argument, but still one that has merit.
I will start with the third. Media that portrays violence in a negative light may educate or propagate its audience into avoiding violent means. Media which exaggerates the negative consequences of violent activities may well have the effect of creating paranoia—even if propaganda itself is wrong. Far better is media which fairly and accurately portrays the truth about violence, which educates the audience about consequences they might not have realized—creating a healthy fear of acting with violence.
It is possible that any correlation between instances of violence in the media and instances of violence in society is a faulty one. Here we have the classic "chicken or egg" conundrum: does more violent media cause a more violent populace, or does a more violent populace demand more violent media? It is fair to consider the idea that widespread economic hardships (whatever the cause) make people more desperate, and that when people are more desperate they commit more crimes and are more violent. Therefore, it is possible that an increase in violent media might be more of a response by the media industry to violent society, than the other way around.
Finally, violent media may also directly or indirectly contribute to an increase in violence in society. In an effort to be more edgy, the media industry may take the lead in portraying violence out of proportion to violence in real life. The variety and prevalence of violent acts may educate or propagate the audience into undertaking violent means. It may inform them of ways to avoid consequences (especially legal ones), and thereby embolden them. It may glorify violence as an acceptable means to an end.
Certainly, facts support the conclusion that there are more instances of murder in a year's worth of media, than in a year's worth of life—especially to one individual. This, of course, depends on the individual's specific diet of media, and the individual's specific tolerance for its content. To make the assumption that a given person is under the influence of violent media is to say that the person is incapable of discerning right from wrong—or in choosing between them. Either that, or it is assumed that the given person is entirely without exposure to any concept of right, whereby being denied the choice.
Both of these assumptions are erroneous. Certainly, there are exceptions—persons with mental illness, who are psychologically incapable of moral judgement, and who may well have gotten that way through birth defect or early abuse. While these are much over-used in fictional media and much over-examined in non-fictional media, blessedly they remain rare. On the other hand, individuals who are otherwise well-adjusted cannot go through life completely devoid of any concept of right.
Media alone can only stand out of the way while a person decides on a violent action—it does not create a desire for violence that is not there to begin with. However, the question remains: "Does violence in the media lead to violence in real life?" Is it a catalyst in allowing violence to escape one's heart? Is it a mere reflection of society? Or is it a tool to be used to discourage violence by airing its repulsiveness and natural consequences?
FEATURED MEDIA: The Sopranos - Unapologetic in its portrayal of mob violence (among other things), this show largely succeeds at painting a picture of the distastefulness of the Mafia world without going to great lengths to disparage it. Its account seems fair, yet I wouldn't want to do it in real life.