It's possible to create a twitter account without a phone number. You'll initially get suspended and get an email stating why. Just reply to that email (open a ticket) and they will approve the account without a phone number.

use a proxy, a new chrome session, and throwaway email address.

But what about the phone number?

most twitter accounts do not have a phone attached to it. you must be doing something to trip up the phone verification thing

They're typically required. Even if you manage to create an account without a phone number, you'll soon start get notifications from Twitter saying that your account is at risk and you have to add your phone for verification (at least that's what happened to a friend of mine who set up an account some time last year)