MOSCOW—The rare defeat of Kremlin-backed candidates in recent regional elections and a slump in President Vladimir Putin’s poll ratings signal rising social discontent among Russians.

Opposition contenders for governor trounced candidates supported by Mr. Putin last month in a region east of Moscow and another in Russia’s Far East, while the Kremlin’s favored candidate dropped out before the runoff in a third region, in Siberia. In another region, on Russia’s Pacific coast, a second runoff vote is planned after the results...