Twitter is a private company, but they create public spaces for communication. This is much like a restaurant. They might be a private company, but their business is a public place. Twitter should not be choosing winners and losers. They should allow free speech, even if they disagree with it.
I personally do not agree with the alt-right but, I think what twitter is doing is very scummy and is creating an echo chamber. However, this is not a violation of the first amendment as they are a private orginazation and are completely allowed to ban whomever they please.
The alt-right has shown it's blatant racisms and intolerance to minorities and other groups in our society. They have promoted hate and that is more of a violation of rights than banning the group from Twitter. This is an act of banning messages of hate from being broadcast to the general public and it is necessary.
No, Twitter is not violating free speech by suspending alt-right accounts. Social media platforms like Facebook and Instagram regulate the content users post, and as a privately-owned platform, Twitter also has that right. The alt-right account owners are free to express themselves through personal outlets like blogs, and are permitted to hold and express whatever beliefs they wish in the public square.
All of the propaganda and "news stories" that come from the alt-right sound like they could be coming from the KKK or some supremacy group for White Men only. I myself am a white man but I can call out bigotry when I see it. That whole group of people don't have a firm grip on reality.