Is it permissible to deposit money in an Islamic bank that engages in prohibited transactions, knowing that other banks do the same?

We cannot issue a ruling on a specific Islamic bank because the strength of Sharia supervision varies among them. To claim that Islamic banks are like usurious ones is false. Usurious banks are based on trading in debts, whereas Islamic banks' primary activity is investing funds through Sharia-compliant methods such as Murabaha and Istisna'a.