Did the Pilgrims experience violence in America?

Yes, they most certainly did, from Native-American tribes. You have to understand that, for centuries, indigenous tribes had occupied America. So when the new settlers came along, taking over land which didn't really belong to them, Native Americans often responded with violence against what they believed were trespassers.
As well as the triple threat of cold, hunger, and disease, the Pilgrims had to endure regular armed attacks on their settlements by irate Indians seeking to drive them back to where they came from. Many of these attacks were totally indiscriminate, leading to the slaughter of countless men, women, and children. For their part, the Pilgrims hit back with even greater savagery, using their superior firepower to subdue Native-Americans, killing them, enslaving them, and destroying their ancient culture.
In such a perfect storm of violence, it was inevitable that only superior levels of violence would prevail—and so it proved, with the Pilgrims' muskets proving more effective than the natives' bows and arrows.