I'll try to answer some of the questions in the follow ups. Sorry for the self reply.
First, governing party cannot change the law in a day. It's more complicated. Not going into political details (everything is politicized in Turkey), there is also a theoretical law issue.
We need such a law. However much more abused, you can see it as the DMCA of Turkey. Turkey (like many European - Old World - countries), adapts a variation of Roman Law, which is in contract to US's Common Law. In US courts do actually make up the law, and decide on their on. In Turkey, there must be a written code for prosecution.
For example in US, if I see my copyrighted work on YouTube, I send a takedown notice, and they comply. (There is a process that everyone knows). In Turkey, I have to go to a PA (prosecuting attorney/state prosecutor), then if he/she finds my complaint worthy, will open a court case. Court will have to decide based on the written law, and send a takedown notice. There is a 30 days period for compliance. This is used to handle cases for copyright/trademark infringement, child pornography, defamation, etc.
Normally this process is more restricted than DMCA, however it all falls apart due to people involved writing and applying the laws (also international websites not caring about Turkish court orders).
Due to my best knowledge, the law in question is adapted from TV broadcast regulations. (This is why I want it completely abolished), and many prosecutors/judges are illiterate on the subject. Furthermore there are some who has an agenda of taking down everything they don't like. So an older angry citizen can go to a court saying YouTube has a video defaming Turkey. Normally this is no big deal (one video in millions). But since the law is broad, the complaint causes the takedown.
The final situation is embarrassing, hope they somehow manage to take down the law itself (and maybe replace it with a much restricted one, as I said unfortunately such a written code is necessary).