This is a great question. Let me give you a few perspectives.
First, while website cannot know for sure what goes up, especially if they give users the right to put things Online. For example, many websites allow comments and within these comments people can even put up photos and videos. Just the massive amount of traffic that websites have prohibits them from monitoring everything that goes on.
Second, however, when the website knows that either criminal activities or suicides are on there websites, they have a responsibility to not only put them down, but also to notify the right people. They may have to notify the police or some other authorities.
In short, there is a responsibility that website have.