What is the ruling on profits I received from a bank that turned out to conduct un-Islamic transactions?

If it is established that the bank's dealings are not Sharia (definition)-compliant, then you must withdraw your money from it. You should then investigate and estimate the proportion of haram (definition) (unlawful) earnings from the profits you have received. Following this, you must donate an amount equivalent to the unlawful portion of those profits for charitable purposes, public welfare, and to those in need. This is because the fundamental principle of an Islamic bank is for its transactions to be Sharia-compliant.