To apply as a refugee, you must register with UNHCR and they will decide where you are settled. If Turkey is your country of citizenship, you will need to leave Turkey and register in another country. It will take years and you don’t decide where you want to be resettled.
If you want to claim asylum, you do that at the border. You need a visitor visa so you need to show proof of funds, travel history, ties to home country etc. to be approved. At the border, you claim asylum with your proof (medical, hosptial, police records etc.) of fear or persecution.