Turkish Airlines Denied Me Boarding Despite Being On Time

Today, my wife, children, and I were supposed to fly from Istanbul Airport to Ordu at 08:05 with Turkish Airlines. We arrived at the airport at 07:00. Although there were available check-in counters, we were directed to the family check-in area since we were traveling as a family. It was crowded, and by the time it was our turn, it was around 07:20–07:25. They told us we were late and refused to let us board. Even though there was still more than half an hour until departure, they didn’t allow us on the plane. One of the staff even told me there were no seats left. How is that possible? Did they say, “Well, there’s still half an hour and some passengers haven’t arrived yet, let’s sell the seat to someone else so it doesn’t go empty”? We were told that to take a flight two days later, we would have to pay an additional ₺9,000. When I asked for a refund instead, they said I’d get only ₺140 back—and I’d have to collect it from a travel agency.