May I be so bold to suggest an answer in response to your rhetorical question whether you should kneel to the protesters in Taksim Square (and other places)?
Without wanting to belittle your own personal achievements in politics, I would point out that greater politicians than you have done it, so why not you?
It would not end the protests. It would not solve any of Turkey’s problems. It would not even solve any problem any Turk might have. It would change nothing in Turkish politics of society. It would not even change or solve anything at all.
But it would do you a world of good. You personally. Seriously.