Republicans say the results in this week’s primary in Texas raise serious doubts about the blue wave that Democrats are counting in the November midterm elections.

While Democratic turnout was high, so was GOP turnout on Tuesday, giving the party confidence that its troops have not been demoralized.

“We kept hearing about this unprecedented Democrat turnout in Texas and the Republicans showed up in droves yesterday,” said RNC Chair Ronna Romney McDaniel.

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The primaries — the first in the nation ahead of November’s general election — set up a number of fascinating races.

Sen. Ted Cruz, a Republican, easily won his primary, capturing 1.3 million of the 1.5 million votes cast. That’s double the number of votes won in the Democratic primary by Rep. Beto O’Rourke, who will be his opponent.

Mr. Cruz said the results puncture “the narrative that a lot of folks in the media want to tell.”

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The general consensus is Mr. O’Rourke is fighting an uphill battle, along with the other Democrats running for statewide office in Texas. Still Mr. Cruz said he isn’t taking anything for granted.

“There is no doubt right now the extreme left is energized. They are angry. They hate the president,” Mr. Cruz said. “We are seeing that in the fundraising numbers for Democrats all across the country. We are seeing that in turnout.”

More than 1 million Democrats voted in the primary, or double the turnout from the last two Democratic Senate primaries in 2012 and 2014. It’s the first time they’ve surpassed the 1 million mark since 2002.

“What’s happening in Texas is part of a national trend,” said Tom Perez, chairman of the Democratic National Committee. “All across the country, Democrats are competing and winning in deep-red states.

Charlie Cook, of the Cook Political Report, a nonpartisan election tracker, said there was a lot of evidence of a Democratic wave heading into Tuesday’s primary and, while it might not cost Mr. Cruz his seat, overall that still seems to be the case.

“It’s the Democratic tidal wave versus the Republican seawall,” Mr. Cook said. “At this point, my bet is that in House, the wave is bigger and stronger than the wall, but in the Senate, my money is on the wall not the wave.”

Democrats believe they have a chance to flip the seats currently held by Reps. Pete Sessions, Will Hurd and John Culberson come November.

But Tuesday also exposed the ongoing divide between Democrats’ more moderate and liberal voices — divisions that dominated the party’s 2016 presidential primary.

In the Houston suburbs, Laura Moser, a prominent anti-Trump activist, overcame attacks from the Democratic National Campaign Committee to finish second. She advances to a runoff in May with Houston attorney Liz Fletcher.

The winner of the primary is set to face off against Mr. Culberson, and Republicans are counting on a bruising primary fight over the next two months to hinder Democrats’ chances.

“Their base is demanding ideological purity and they are forcing these candidates farther to the left to adopt positions like single payer health care that just aren’t tenable with a broader electorate, particularly in Texas,” said Jesse Hunt, spokesman for the National Republican Congressional Committee.

Another liberal favorite, Gina Ortiz Jones, a lesbian and Iraq war veteran, advanced to the runoff in a western Texas district, putting her a step closer to a showdown with Mr. Hurd.

Jim Dean, chair of Democracy for America, said Democrats can win if they nominate candidates cut from the same cloth as Mrs. Moser and Ms. Jones.

“We’ve got a long road to November, but the wins we saw last night in the Lone Star State, as well as the runoff primaries some of those victories set-up, strongly suggest momentum is building behind a nationwide Blue Tsunami — if Democrats get the grassroots-inspiring candidates we need to generate the waves,” Mr. Dean said.