Is it permissible to buy a car in installments from an Islamic bank, authorizing the bank to purchase, choose, and determine the profit based on the payment period and the value of the commodity, and to pay in Turkish Lira, taking into account the foreign exchange rate on the day of payment?

What you mentioned about the transaction is not enough to rule on it, due to an incomplete understanding of it. However, among the conditions for its permissibility are that the bank must fully own the commodity and take possession of it before selling it to the one who orders its purchase, and there must be no condition stipulating a penalty for late payment of installments. Organized Tawarruq is impermissible, and it is not permissible to agree in the contract to repay the debt in another currency.