Not sure what to do...

At what point do you let your child investigate (search) for things online on their own and not worry about the "trouble" out there waiting to find them? Anna has been looking at and researching ok things online, but she was told something at "school" over a year ago that she had to research on Netflix and we do NOT in any way want her looking at this content or anything like it again. So, now the question is, how much freedom do we give her online. We've blocked a lot off her iPod Touch, but she is not always with us. Sigh. Just not sure where to go with this. She hasn't been at a public school in a year. Well, almost and she is just now remembering things that were said to her there and trying to find out for sure if they are real. Just scary content that she doesn't need to deal with.