Turkish Airlines Overbooked Flight Resulting in Denied Boarding

I purchased tickets from Turkish Airlines for myself and my brother. We were informed that the flight was overbooked and there were no seats available, so we were not given tickets. At the last minute, I was given a ticket, but my brother was not. When I reached the gate, the staff member allowed the person before me to board with a 5-second difference, even though they saw me running. Despite assisting a lady, we were not allowed to board the plane. The staff claimed there were no seats left and turned us away. Upon returning, I was treated as if I had missed the flight and was told my ticket was void. I had to purchase a new ticket with only a tax refund. The gate staff mentioned that they boarded someone else by claiming we were late. We were told that we should have been given seats on the next flights and even had compensation rights; however, I was only able to get a ticket excluding taxes. Please take the necessary actions regarding this issue.