Turkish Airlines Disrespectful Staff

I had a flight from Antalya to Trabzon that included a stopover in Istanbul. Unfortunately, the Antalya-Istanbul flight was delayed, and I requested assistance from the airport staff to avoid missing my connecting flight. They let me board the 15:55 flight (which is visible in their system). We also scheduled our connecting flight from Istanbul to Trabzon for 19:20. When we arrived in Istanbul, we were shocked to learn that we were not permitted to board the Trabzon flight! Because our initial flight was delayed, the system apparently automatically postponed our connection without informing us! We didn't make any changes, so why wasn't the information in their system updated? We explained the situation several times at Gate G3-J at 19:00, but the staff was dismissive and irresponsible. The person in charge even gave our seats to someone else and then left without saying anything! They didn't give a damn about us. Turkish Airlines was completely unprofessional in leaving us stranded with a small child until 22:30 due to their incompetence.