It's a well established fact that conservatives are, at a minimum, just as guilty of hostility to ideas they don't like as liberals.
Consider Megyn Kelly, a celebrated conservative journalist whose career was completely destroyed simply because she asked Trump some tough questions.
Consider Tomi Lahren who was a cherished conservative icon until she came out in support of abortion based on her conservative principles of individual freedom... following that expression of free speech she was squeezed out of her job and destroyed in the conservative press.
Consider Gretchen Carlson who was humiliated and dragged through the mud in conservative circles after she spoke out about her own experiences with sexual harassment in the workplace.
Consider John McCain who is regularly blasted as a RINO because he occasionally takes an action that is not in complete lockstep with the party line.
Consider Glenn Beck who was on the receiving end of boycotts and death threats because early on, he spoke out against Trump in favor of the eminently conservative Ted Cruz!
Consider Ted Cruz himself who was literally booed off stage during the RNC because he dared to instruct the audience to "vote with your conscience" instead of blindly praising Trump.
The list goes on and on, so I have to reject your assertion that "suppressing free speech is 90% leftist" as utter ignorance.
Well there's parallels for a lot of groups.
Some cops find new careers as politicians and then threaten reporters who ask questions they don't want to answer because that's what cops often do (I know I'm over-generalizing).
Edit: Also who could forget the brouhaha around the Dixie Chicks after they said something bad about George W. Bush [2]. That incident foreshadowed the kind of tactics that are common on the left now.
[1] http://www.politico.com/blogs/on-media/2016/02/donald-trump-...
[2] https://en.wikipedia.org/wiki/Dixie_Chicks:_Shut_Up_and_Sing