There is a fine line between being a caring parent, and being overprotective. Where that line is drawn is a tough call to make. For example, let’s look at the norms of what is acceptable to show on broadcast TV. Agree or disagree, the standards in various countries are a reflection on what is generally accepted as OK to show to the general public.
In the USA, massive amounts of violence are OK to show, but sex and nudity are taboo. In many other parts of the world, the opposite is true: sex and nudity are OK, but violence is not. The attitudes present in the USA can be traced back to the “sex is bad” repression that has been advanced by Christian religious institutions. The negativity towards violence in other countries is a more natural expression of the fact that hurting people is not something that should be viewed so often that people become desensitized by it.
So what does this mean for you as a parent? Just don’t be completely ignorant of the environment and the fact that sex and nudity are natural, while violence is not. If you’re going to be protective about one area over the other, follow the example set by Europe rather than the USA.