In the election races across the country Tuesday, Houston voters rejected a transgender “bathroom bill,” Ohioans torched legal marijuana, and Republicans captured both gubernatorial contest races at stake.

With 30 percent of precincts counted, The Associated Press declared that Houston’s Proposition 1 had gone down to defeat. The measure was taking a beating with 62 percent of voters against the measure and 38 percent in favor.

The vote capped a pitched 18-month battle between Houston pastors and Mayor Annise Parker, the first openly lesbian mayor of a major city, who had championed the ordinance allowing people to use restrooms, showers and other public accommodations based on their gender identity, not their biological sex.

In Kentucky, Republican businessman Matt Bevin defeated Democratic Attorney General Jack Conway by 53 to 44 percent, with independent Drew Curtis taking 4 percent in the race to succeed term-limited Democratic Gov. Steve Beshear.

Meanwhile, Mississippi Gov. Phil Bryant easily won a second term over Robert Gray, a Democrat who spent just $3,000 in a token campaign in the overwhelmingly Republican state.

Ohio voters overwhelmingly rejected Issue 3, which would have legalized recreational marijuana for adults 21 and over, handing the pro-pot movement its first defeat since Colorado and Washington approved legalization measures in 2012.

In Ohio, however, there was a twist: The proposal would have also created a monopoly on cultivation for 10 investors, prompting opponents to denounce the measure as an example of “weed greed.”

“When it comes to the broader debate about legalizing marijuana, the defeat of Issue 3 won’t be a case of ‘as Ohio goes, so goes the nation,’” said Tom Angell, chairman of Marijuana Majority, in a statement.

Unlike Ohio, the four states — Colorado, Washington, Oregon and Alaska — that have approved recreational pot for adults began with medical marijuana.

“This was about a flawed measure and a campaign that didn’t represent what voters want,” Mr. Angell said.

Mr. Bevin, a Kentucky business owner who had challenged Senate Majority Leader Mitch McConnell in the 2014 GOP primary, ran as a political outsider with strong tea party support. He won the GOP gubernatorial primary in a crowded Republican field by just 83 votes.

In his victory speech, Mr. Bevin issued a clarion call for unity, praising Mr. Conway and saying “this is our opportunity to come together as one. We have much work ahead of us.”

It is time “to get the overalls on, get the boots on and get out of bed,” he said.

His running mate — Lt. Gov.-elect Jenean Hampton, who introduced him Tuesday night — became the first black person to win a statewide race in Kentucky.

Mr. Conway made a concession and congratulations call before 9 p.m., telling supporters at the Frankfort Convention Center that he had told his rival that he “remained positive about moving this state forward, and that if he ever needed any assistance, that this Democrat was at his disposal.”

The result came as a bit of an upset because every poll had shown Mr. Conway ahead of Mr. Bevin, albeit only slightly in recent weeks. His election nearly completes Kentucky’s shift from Democratic to Republican over the past few decades as GOP candidates gain ground in statewide races.

Running as a conservative Christian, Mr. Bevin was known for his unswerving support of Rowan County Clerk Kim Davis, who drew national attention over her refusal to issue marriage licenses to gay couples.

By contrast, Mr. Conway, when he was attorney general and a year before the Supreme Court’s ruling, decided not to appeal a federal judge’s order that the state marry same-sex couples, effectively imposing gay marriage on the state by default.

The Bevin campaign hammered Mr. Conway over his support for President Obama, with ads saying that the Democratic candidate was in favor of Obamacare, gun control and abortion but against coal, “just like Obama.”

The American Principles Project declared in a news release that social issues and candidates who support traditional values had won big.

“Bevin’s upset victory in Kentucky is a victory for Kim Davis, a rebuke to those who say social issues hurt the GOP, and a call for Republicans nationally to support the First Amendment Defense Act,” said the American Principles Project’s Maggie Gallagher in a statement.

Mississippi Attorney General Jim Hood was also leading in his bid for a fourth term. The Democrat’s race was especially significant in that he is believed to be the only Democrat still holding statewide elected office in the Deep South.

Whatever happens in Mississippi, Louisiana could change that later this month. That state already had its so-called “jungle primary” for governor last month, narrowing the field to the top two vote-getters, Democratic candidate John Bel Edwards and Republican David Vitter. They now square off on Nov. 21.

More than 300 cities held mayoral elections, the biggest being Houston and Philadelphia, America’s fourth- and fifth-largest cities.

In Houston seven candidates face off in a nonpartisan race to succeed Mayor Annise Parker, who pushed the transgender ordinance but is term-limited herself.

In overwhelmingly Democratic Philadelphia, former city council member Jim Kenney easily defeated a Republican businessman to succeed term-limited Mayor Michael Nutter.

• Stephen Dinan and Victor Morton contributed to this article, which is based in part on wire service reports.

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