I started reading when I was 3 and constantly read whatever my mom had laying around, regardless of its age-appropriateness. We never had any video games so that wasn't an issue, but there weren't many movies or television shows our mom expressly forbade if they were on and there were adults present. We were also allowed to buy whatever CDs we wanted, but neither my sister nor I never really listened to rap or metal and I think if we had my mother would have put the kibosh on those purchases out of her own prejudices.
For me it resulted in a broad vocabulary, a slightly worrisome interest in the paranormal and the macabre, and my tendency to sleep with the covers over my head. There were many things that came up in what I read or watched that I just didn't understand until I was older, and I think that's the case with a lot of children, but the (sometimes justified) fear isn't so much that they'll comprehend what they're seeing but rather that they'll try to imitate it without understanding what they're doing.
Really it's the responsibility of every parent to know their child and figure out what will and will not be appropriate for them. Ratings systems are useful in that regard, and resources that discuss what kind of content is in any given media. That means I can do my research before making a purchase; I don't know why you wouldn't bother to at least look things up before spending a fortune on taking your kids to the movies and then get upset when it turns out there's something in it you don't consider appropriate for them, or they wind up getting scared.
Then again, kids change, especially little ones: we played Skyrim around LD from the time he was a few months old, but now that he has a different grasp on the world around him he tends to get scared when a dragon appears on screen. That's part of being a parent, too: continually evaluating and reassessing your child's needs.
"And we are all told from the moment we open our eyes, that everyone is entitled to his or her opinion. Well, thatís horsepuckey, of course. We are not entitled to our opinions; we are entitled to our informed opinions. Without research, without background, without understanding, itís nothing. Itís just bibble-babble. Itís like a fart in a wind tunnel, folks." --Harlan Ellison