The most common misconception about martial arts is that they are violent. And while martial arts are violent in nature, the best martial arts, those that are most widely practised, are not based on violence but discipline. Though they have a violent edge. But the best martial arts are not violent.
In order to understand this, one must first understand that violence is not the only thing that martial arts teach. Martial arts teach fighting, but not in a "fighting" manner. Nor is it the only thing that sports teach. And though sports are primarily physical, they do not teach fighting in a "fighting" manner.
Though self-defense is a big part of martial arts, and is a teaching in itself, it is not the primary teaching. There are sports that emphasize this more than others. And while we see this taught in movies, we see this taught in real-world situations too.