Turkish no baggage allowance on award flight

Update-

I contacted United several times about the refund. After denying my claims a few times they decided to send my complaint to TK.

Now TK has come up with the most outrageous story: apparently I went to the check in counter with two bags, they checked one for free and the other one weighed over 80kgs were charged! I don’t even to know how to respond to this most outrageous lie! I did check in 4 bags in total for 3 passengers and didn’t get any free allowance for any of us. In total we checked in 86kg and were charged for every single kilo of that. I’m beyond frustrated that they are playing this nasty now.

So I bought 2 award flights (2 adults and 1 infant on lap) on Turkish from LHR to IST to South Asia through United. I contacted Turkish on the phone to confirm the allowance for my infant and was told each adult gets 40kg and the infant gets 10kg. United also told me that my baggage allowance will follow Turkish protocols which according to TK website is 40kg checked bags- weight based concept for business class.

Anyhow fast forward to a few days ago- when I tried to check in at The airport I was told that I have zero baggage allowance! The Turkish agent advised me to talk to United ticket desk since they have issued my ticket. Went over to speak with United ticket desk and they told me that they can see my baggage allowances online but not sure why Turkish can’t see it. United ticket desk agent even put a note on the reservation mentioning my luggage allowance and told me to pay for my bags if Turkish still doesn’t honor it and United will reimburse me later. Later a United agent on texting app also advised me to pay for the bags now to claim reimbursement through United later. I have the screen record of the messages. At one point Turkish even told me that my reservation got voided or canceled so I had to go back to United ticket desk again to get the tickets reissued.

Long story short this whole ordeal took close to 2 hours to resolve and I ended up paying close to 2000 GBP for our bags ( per kg 24 GBP or something) :(.

I did some digging afterwards and it seems like a known issue with TK. What should be the best course of action now to get the baggage charges refunded? Should I just do a chargeback on my credit card using the proof that united said they would reimburse me and my original ticket issued by united that showed the baggage allowance? Or try getting it reimbursed through united first?