Turkish Airlines Flight Departed Early

On my 15:40 Istanbul-Gaziantep flight from Istanbul New Airport with Turkish Airlines, I was told that the plane had departed 15 minutes earlier when I arrived at the gate at 15:20, rendering my ticket invalid. When I went to the counter with my child in my arms, I was informed that there was no audible announcement and questioned how the 15:40 flight could depart at 15:05. The response I received was, 'This is how the system works, you should have been there earlier and checked the screen.' I wanted to purchase a new ticket to resolve my issue, but I was met with a rude response saying, 'You need to buy a new ticket from scratch, this is your fault.' I was left in a difficult situation with my small child. Additionally, my luggage was damaged, completely torn apart, although they did provide a new suitcase as compensation. My child and I were left stranded and lost our money. I request that this issue be resolved or that I receive a refund.