Turkish government's intention here is getting content that criticise them removed, by asking to representative in Turkey who by the way must be a Turkish citizen, not just a person from any nationality who can speak Turkish. So that, they can apply Turkish law to this person if they would like to. If Youtube doesn't remove the content, possibly as they did with TV they are going to fine them.
I don't see this model working for long time and expecting most of the companies to pull their representatives soon, if they open the door of every country getting content removed however they like, I'm sure there will be other countries who would want that and it would be terrible for freedom of speech.