How does forcing them to use a different communication medium stop them from spreading ideas you disagree with?
If all the newspapers and TV stations in the world refuse to run a news story, then it'll prevent the story from spreading. It doesn't entirely stop it, obviously. Even before the Internet, there would have still been word of mouth. Still, it prevents it from spreading to the extent that it would have otherwise.
It seems to me that giving them the allure of being the 'stuff THEY don't want you to see' only helps promote it, instead.
You're conflating two things. You're talking about something like the Streisand Effect, where trying to hide information paradoxically causes it to spread. That can happen, although if reputable sources of information refuse to acknowledge it, it might still be relegated to the status of rumor. I'm talking about a different thing, which is more about whether credence and credibility are given to speech. Racism, for example, isn't a secret that people are curious about. No one is sitting at home thinking, "I heard something about this white supremacism. No one has ever been willing to advocate lynching, so the idea is so much more alluring now!"
It's more like, there are various people who are racist to varying degrees and in different ways. That's already in their lives. If the people around you who are credible members of the group you perceive as belonging to are all lynching people, talking about lynching, and advocating lynching, then there's a much greater chance that you'll end up lynching someone. If the suggestion of lynching elicits a response of "Hell no. That's fucked up. What's wrong with you?!" then you're less likely to lynch anyone. That's just how people work. If services like Twitter promote and amplify hate-speech, you're going to end up with more people thinking it's a normal and acceptable thing. If Twitter bans it and sends the message that it's unacceptable, then its prevalence lessens.
And yes, I know there will still be some backlash. There are white supremacists who are going to be irate any time you imply that white supremacy is not acceptable. There are some occasional assholes who will say the exact thing that that they think will be most offensive and get them the most attention. However, ultimately most people will generally adopt the social mores of whatever group they perceive themselves to be a part of. A responsible member of society tries to avoid and discourage horrible behavior and speech in order to encourage better social mores.