Believe it or believe it not, but many parents really don't have much of an idea when it comes to the internet. I know, because I advise many parents about IT matters - not in their capacity as parents, but when they're at work, running their companies. many's the time I've been asked to make a visit to a client's house to 'sort out the kids' computers' or to teach the boss how to use Microsoft Word, at home, where his staff can't glimpse his lack of expertise.
When I make these visits I'm repeatedly surprised to discover how many children are using high-powered, well specified machines in their rooms, and how many parents just don't have the faintest clue about how it all works. They provide the children with broadband access, and although some of them make a token request that I "make sure they can't see the wrong kind of stuff", most of them don't even go that far - they assume that because all their childrens' friends have internet access, and the school has its own website, all must be well. They've heard of kids getting into trouble via the internet, but they're not sure how, and in any case, "it won't happen to my children, they're far too sensible."
Most of the time they're right of course - relatively few children come to any harm because of their use of the internet, and it's worth bearing that in mind. I'm not aware of the facts surrouinding the case you mention, but my gut feeling is that if a child is " driven to attempted suicide by her peers!" it's highly unlikely to have been just the internet that tipped her over the edge. It's far more likely to have been the peer-group pressure, and that can happen without the internet ever being involved.
It's easy to reach for the internet as being the culprit in such cases, but in reality it's rarely that simple. Of course there's internet content that is totally unsuitable for viewing by minors, and of course there's a need for parental influence, but let's be careful about a knee-jerk response. Anyone who has been the parent of an intelligent, spirited child with a highly-developed sense of independence will know that it isn't just a question of saying 'No' - you have to be prepared to provide a little more than that by way of a rationale.
The days of the master of the house ruling with a rod of iron, and "never mind why, you'll do as you're told" are long gone - thank goodness.