The government have complete control of all ISPs and mobile network operators in Turkey. They don't own them, but you can't give people Internet access without joining the government censorship/logging program.
As she was a teacher she would be employed by the government, and they just fire everyone employed in the government automatically when their names show up on the blacklist. I suspected firing of family members and friends that are connected to falsely accused person is done manually.
Also it doesn't really matter if the person is employed by the government or not. No employer would dare to keep anyone that is blacklisted as the employer would probably be accused themselves of helping a terrorist if they kept them. Plus, obviously, most big corps in Turkey are run by Erdoğans friends and family directly and indirectly.
Probably that's just the journalist words as I don't think they found or even looked for "traces of the app" on the phone. Most likely the government got a log of the DNS lookups from the ISP and made their arrests based on this.