"The internet is difficult to regulate. Neo-nazis use it to co-ordinate their activities unchecked, and to spread as much hate-filled material through the net as possible. You can't make accessing it impossible, but you can make accessing it illegal. You can make it illegal to spread false propaganda that's only intended to harm people and cause harm. " [Boldface mine]
What is "false"?
What is "propaganda" and what are facts?
What is intended to "cause harm"?
If society could define thse concepts universally, your solution might work. Unfortunately, to take some examples from the U.S., those who support the right of a woman to have an abortion could be assailed by the Christian Right for putting out "false propaganda that's only intended to harm people and cause harm." They could say the same thing about evolution. Conversely, humanists could lay these same charges against religious thinking.
One person's "falsehood" is another person's "truth." As long as we cannot agree on standards such as these, it will always be dangerous to make certain types of statements illegal.