At least 11 conservative state lawmakers in Kansas, including many allied with Republican Gov. Sam Brownback, were ousted in a primary election Tuesday, in a rebuke of tea-party advocates who had pushed through tax cuts that many viewed as endangering public-school funding.

Among those who lost their seats were Republican Senate Majority Leader Terry Bruce and five other conservative senators who had been allies of the governor. Five conservative House members from the Kansas City suburbs in Johnson County also lost their...