Turkish Airlines Flight Overbooking Issue on TK 2223

I want to highlight an issue with Turkish Airlines that left me in a difficult situation. My name is I***. Four months ago, I purchased a ticket from Istanbul to Gaziantep to visit my dear parents. I arrived in Gaziantep without any issues from Istanbul Airport.
Today, on April 13, 2024, I arrived at the airport 1 hour and 40 minutes before my flight. With reservation code Rv83h5 for flight TK 2223, I was informed that ten extra tickets were sold, the aircraft type had been changed, and everyone had checked in online, leaving no available seats. Despite having a ticket and being at the airport 1 hour and 30 minutes before departure, I was not allowed to board the flight I booked four months ago. Along with me, ten other passengers faced the same issue. The reason was that Turkish Airlines had overbooked by ten tickets and changed the aircraft.
We had planned to check in at the airport. My 2.5-year-old daughter and my wife received tickets, but my 5-year-old son and I were given standby tickets and sent to the gate. We were told that if there were any no-shows after the last call, we would be allowed to board. Of course, there were no available seats, and we were not allowed to board. When we asked what would happen next, they said we would receive compensation and a new ticket. At the ticketing counter, they mentioned a compensation of €100 each. Meanwhile, my 5.5-year-old son was crying and agitated because he was separated from his mother. Additionally, all of his belongings were in the suitcase that had already been loaded onto the plane. We later learned that our luggage would arrive in Gaziantep on the 22:30 flight.
Despite explaining that I am a doctor with surgeries scheduled for Monday and that I am a government employee without the possibility of taking leave, the earliest ticket they offered was for Thursday morning at 8:20.
I tried using my connections to find a bus ticket, but due to the upcoming holiday, all buses were fully booked. After involving many people, I managed to find a ticket from Adana. I am now taking a taxi for the 2.5-hour journey from Gaziantep to Adana, costing me ₺4000, to catch the bus.
I am extremely dissatisfied with Turkish Airlines for this situation. I will never travel with Turkish Airlines again and will pursue my rights to the fullest extent.