Online travel bookings

I have made previous bookings with Travelstart and was pleased with the service. In June 2023 I once again visited their website to check on cheap flights for my travel plans. After searching for flights on their website, I found a cheaper flight on another website, I then left Tavelstart website and made a booking with another company. On the same day I noticed that there was a deduction on my bank account for travelstart. I immediately notified them of the unauthorized payment. I have not given them my bank details or a confirmation with OTP as done always when I do online transactions .After numerous calls and even a physical visit to their offices, they will not refund at all. I can use the ticket on future flights valid for one year. The fact that I made previous bookings with them and I have a profile on their website, was stated as the reason the transaction went through. My daughter is going on honeymoon in October and I am cautions to go on their website to help them check for cheap flights.