You can not legally work untill you have your work permit signed by the ministry, period.
The SSK people can come around anytime and make sure that everyone working there is paying social security. Anyone that is not paying will end up being fined and also the company getting fined aswell.
I know there are people working here illegally for years but it still does not make it legal and it does not matter who you know when these guys come around and they have been snooping around istanbul for the last few months doing snap inspections.
Any company, no matter if its a school or a corporation, HAS to apply for your work permit for any foreigner they employer by law.
As for english time, well no need to go there as their reputation speaks enough for itself.