90% of the violence occurs during the Vietnam chapter, explosions, gun shots, dead bodies, etc. other than that nothing else happens except (non-graphic) punching and bullying.
A man slaps Jenny and Forrest beats him up briefly.
Forrest finds and saves plenty of soldiers in the jungle. The men have blood on them. One of the men hurts his legs so badly to the point that they had to be amputated.
Real-life footage of a man being gunned down and blood seeping through his shirt is shown for a few seconds.
Forrest runs to a car and he starts punching the driver but he is stopped by Jenny. The reason why he did it because he thinks the driver was hurting Jenny and she claims he wasn't.
Forrest has to carry the injured out of the jungle in a Vietnam war scene and people are seen with bloody wounds on there body caused by gun shots and explosions.
A woman nearly commits suicide by jumping from a building.
Bullies insult, throw rocks at, and chase another boy (one rock hits him in the head and we see a little bit of blood). Later, the boy is shown older and he is again chased by bullies in a car. He manages to escape from the bullies both times, unharmed the second time.
Some other more violent historical events are also shown, including newsreels of George Wallace getting shot (we see some blood) and assassination attempts on Gerald Ford and Ronald Reagan. The assassinations of John F. Kennedy, Bobby Kennedy, and John Lennon are also referenced, but not shown.
Several fistfights occur throughout the movie that result in bloody noses, black eyes, and other minor injuries. One such fight ensues from a man's violent reaction to a woman getting slapped across the face and knocked to the ground.