WARSAW—Poland is giving the government sweeping powers over its judiciary, a move that is rattling European Union leaders and U.S. diplomats but hasn’t shifted voters much at home.

Opinion polls show steady support for Law and Justice, Poland’s ruling faction, since the 2015 election. Prior to the Senate passing a law early Saturday giving the ruling party the right to dismiss the entire Supreme Court, a survey by Warsaw-based research institution Ibris showed that 37% of Poles continue to back the party, a 16-point lead...