For the Republicans who have run Texas state government for years, trying to undermine the Democrats who lead the state’s largest cities has been a blood sport for years. They have sought to overrule local officials on disputes involving everything from pandemic restrictions and plastic bag bans to protections for immigrants. But this week, the collapse of the state’s power grid gave Democrats a chance to turn the tables. With the state reeling from a rare winter storm that caused widespread power outages, Democrats have mobilized public anger over the Republicans’ oversight of the energy industry, opening a new front in their battle to erode the party’s dominance of every statewide office and both chambers of the legislature.

While Democrats have made important inroads in recent election cycles, Texas Republicans have staved off the kind of game-changing gains that flipped states like Arizona and Georgia.

“Those in the Legislature and those in the executive branch of government have been spending too much time trying to run cities and counties and not enough tending to state issues,” said Sylvester Turner, the Democratic mayor of Houston, the largest city in Texas and the fourth-largest in the country. “And now it’s coming back to bite them. Before you can try to run my house, you’ve got to make sure you’re running your own.”

Such appraisals come at a time when Greg Abbott, the Republican governor, was already under fire for his handling of the pandemic. Even before thawing freezers forced health officials in Houston to scramble this week to administer thousands of vaccine doses, Latino leaders in hard-hit South Texas were pleading with Mr. Abbott to allow city officials along the border to put into place stricter mitigation measures.