Right, let's take this bit by bit.
The way a large majority of the left behave towards those who step out of line from their orthodoxy is very, very similar.
Let's leave the weird stuff about the Nazis rising to power using No Platforming techniques, and look at this statement. It's bollocks. Who are this 'large majority'? Corbynistas within the Labour Party? Trots? I've been on the Left my whole life, and I don't recognise it. You could argue that New Labour attempted to introduce some authoritarian policies such as ID cards, I suppose. Who has introduced the recent 'Snoopers Charter', though? This Tory government. Who are trying to fiddle constituency boundaries for their own benefit? The very same. There are authoritarians and Libertarians on both political wings; try looking at the Political Compass.
I'll try and explain it more simply then. Every group in history which has acted as an oppressive force which has silenced their opponents, denied their right to hold different views and sometimes even exist have believed that they have a right to do that because their views are inherently right so no opposition should be tolerated. And their opposition are painted as inherently bad and evil who need stamping out and their views oppressing to protect the views of the repressing group who believe that they alone hold the correct and right views. The left does that today and justifies its attacks on opponents by portraying themselves as inherently 'good' and 'correct' which gives them a right to attack those who dissent from them.
Both sides do this. People who hold strong political beliefs; in fact, beliefs of any stripe, think that they are correct.
But it never bloody is. If the only way you can promote your ideology is by silencing your opposition that's because it's a pretty shit ideology and you'll never be able to convince enough people to support it if you have even a halfway decent opposition.
There's a lot of paranoia coming over in your posts. Who are in power here and in the US? The Right are triumphant at the moment; arguably the extreme right in the US. Who are being silenced here? The 'Alt-Right' are bombarding people with death threats and calling anyone who disagrees 'snowflakes'. Gina Miller, Anna Soubry, and those opposing Brexit are being harassed and silenced by our right wing print media. The judiciary are being called 'enemies of the people' by the Daily Mail. It's less than a year since Jo Cox was murdered by a fascist.
Your side are winning. Your side always tend to, because that's inevitably where the power resides. So why are you right wingers so fucking angry all the time? You don't get to silence us by stamping your feet. You are doing exactly what that old bastard Goebbels said; 'Accuse the other side of that which you are guilty'.