Japan’s Liberal Democratic Party, which has led the country for most of the last seven decades, appeared to be heading toward a major defeat in parliamentary elections on Sunday, as exit polls showed voters shifting to opposition parties that included new right-wing populist groups.

The vote for the less powerful Upper House of the bicameral Diet, as Japan’s Parliament is called, was being closely watched as a referendum on the governing party and its leader, Prime Minister Shigeru Ishiba. Under Mr. Ishiba, the Liberal Democrats have struggled to deal with a variety of issues, including the rising price of rice, a backlash against increased immigration and the threat of tariffs from Washington.

After voting ended, exit polls showed the Liberal Democrats and their coalition partner unlikely to hold on to the 50 seats of their 66 up for re-election that they needed to retain control of the chamber. As many as two-thirds of the 124 seats up for re-election appeared set to go to a host of opposition parties spread across the political spectrum.

Even as ballots were still being counted in dozens of close-fought races, powerful members of the governing party were already calling for Mr. Ishiba to step down to take responsibility for what seemed to be a poor showing. Taro Aso, a former deputy prime minister, said he “couldn’t accept” Mr. Ishiba staying on as prime minister, Japan’s TV Asahi reported.

While Mr. Ishiba said in a television interview on Sunday that his party had not done well, he said he wanted to continue as leader because he still had “important duties to fulfill for the nation.”