Who bears the responsibility for any malfunction or damage to the car agreed upon with the bank, before its ownership is transferred to me?

If the sale occurred between you and the bank, such that the car was owned by the bank and then it sold it to you and you took possession of it, then you are responsible for it, even if ownership has not been transferred to your name. This is because transferring ownership is a legal and documentation procedure, and not a condition for the contract of sale or for responsibility. However, if what happened was merely a promise to sell and the sale contract was not concluded, then the responsibility for it lies with the bank, because it has not entered into your possession and you have not taken hold of it.