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DES MOINES, Iowa—After a day of confusion and recriminations over a technical mishap that snarled the vote count, the first wave of delayed results from Iowa’s presidential caucuses showed Pete Buttigieg and Sen. Bernie Sanders at the top of the Democratic pack.

The early numbers, representing nearly three-quarters of precincts, suggested a possible split verdict that could have one candidate winning the popular vote and another securing the most delegates, the measure state and national party officials have said should determine the victor.

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A close finish between Mr. Buttigieg, the former mayor of South Bend, Ind., and Mr. Sanders, of Vermont, would also underscore the larger Democratic primary battle between progressives and more-moderate candidates in a field still totaling close to a dozen.

More broadly, the muddled results and fallout from the chaotic count stoked tensions among the candidates and concerns among party officials that the process is doing little to find a strong nominee who can unite the party.

Former Vice President Joe Biden, a longtime front-runner who has built his campaign on the idea that he is that candidate, appeared headed for a worse-than-expected finish. He is competing with Mr. Buttigieg and Sen. Amy Klobuchar of Minnesota among the more moderate candidates, and the Iowa results suggest he was beaten by someone in Mr. Buttigieg who was barely known a year ago and who has never governed anything larger than a city of about 100,000 people.

The moderate contenders also face a growing challenge from billionaire Michael Bloomberg, who wasn’t on the ballot but who has risen in polls after spending several hundred million dollars on advertising. He said on Tuesday that he would further step up spending and hiring.

Sen. Elizabeth Warren of Massachusetts, representing the more liberal wing of the party, was below Mr. Buttigieg in the early results but above Mr. Biden.

Ms. Klobuchar, who frequently says on the campaign trail that she has never lost an election since fourth grade, appeared headed for a fifth-place finish in Iowa.

Though the state’s population is small and not very diverse, Iowa’s caucuses have had a reasonably good record in recent history of picking the eventual winner of the Democratic nomination: Al Gore in 2000, John Kerry in 2004, Barack Obama in 2008 and Hillary Clinton in 2016. The caucuses have been less predictive among Republicans in recent election cycles.

There are indications the Democratic Party is unlikely to resolve its debate over candidates and vision soon. All four of the top candidates heading into the caucuses had campaign war chests large enough to sustain them through February, including the New Hampshire primary next week.

A self-described democratic socialist, Mr. Sanders has recently shown strength in national polls. That has triggered concerns among centrist Democrats who worry President Trump, a Republican, would use Mr. Sanders’s political identity to damage the party’s prospects in November.

The release of the results followed a brief news conference Tuesday afternoon with Iowa Democratic Party Chairman Troy Price, who said the nearly- one-day delay for even partial results was unacceptable and happened because of “multiple reporting challenges” as about 1,700 precincts began to feed results into state party headquarters.

Mr. Price declined to say when all of the state’s results would be made public. “We’re going to take the time we need,” he said.

The state party is collecting paper records completed during the precinct meetings. It has attributed the holdup to a software glitch that triggered inconsistencies in data obtained from a mobile app used to report vote tallies.

“We know this data is accurate,” Mr. Price said. “We also have a paper trail.”

With 71% of precincts reporting, Mr. Buttigieg had the largest share of delegates to the state party convention, while Mr. Sanders had the second most. But Mr. Sanders led in the raw vote so far, followed by Mr. Buttigieg.

Ms. Warren and Mr. Biden were third and fourth respectively in both counts.

Mr. Buttigieg, a 38-year-old openly gay man who is the youngest in the race, could see a bump following what appears to be the best showing of any moderate. He has positioned himself as a voice for generational change—most of his top opponents are septuagenarians—who is willing to work across the political aisle.

In New Hampshire, Mr. Buttigieg has also recorded strong support in polls, but he faces a stiffer challenge in the more ethnically diverse states of Nevada and South Carolina this month, as he has so far received virtually no support in polls from black and Latino voters.

The muddled reporting gave all the top candidates room to frame the result as a win. Speaking to reporters Tuesday evening in New Hampshire after the release of some Iowa results, Mr. Buttigieg was asked what it would mean to his candidacy if he ends up finishing in second or third place. “It still amounts to an astonishing victory for our vision,” he said.

In a statement, a senior adviser to Mr. Sanders called the early numbers gratifying. “In the partial data released so far, it’s clear that in the first and second round more people voted for Bernie than any other candidate in the field,” Jeff Weaver said.

Ms. Warren said at her first New Hampshire event on Tuesday, “We know that three of us will be dividing up most of the delegates coming out of Iowa. I’m feeling good.”

Mr. Biden, during an appearance in Nashua, N.H., expressed optimism after arriving from Iowa. “It’s like my mother said, ‘Hope springs eternal,’” Mr. Biden said. “We’re not giving up.”

State party officials in Nevada, where the second caucus of the nominating season will be held, on Feb. 22, pledged that they wouldn’t experience the same failures as Iowa.

“We will not be employing the same app or vendor used in the Iowa caucus,” Nevada State Democratic Party Chair William McCurdy said in a statement. “We had already developed a series of backups and redundant reporting systems, and are currently evaluating the best path forward.”

Democratic National Committee Chairman Tom Perez said the app that misfired in Iowa wouldn’t be used in any other state, either.

“What happened last night should never happen again,” Mr. Perez said. “We have staff working around the clock to assist the Iowa Democratic Party to ensure that all votes are counted.”

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Meanwhile, Mr. Trump pounced on the uncertainty, calling the Democratic caucuses an “unmitigated disaster” in a Tuesday morning tweet, ahead of his State of the Union address that evening.

The reporting was complicated in this year’s caucuses by additional data. In response to calls for greater transparency following the close 2016 contest between Mr. Sanders and Mrs. Clinton, the state party was to release for the first time the raw total for votes from both rounds of voting, along with the traditional share of delegates won.

Mr. Sanders has been ahead of his rivals in recent state polls in New Hampshire, but Ms. Warren, Mr. Biden and Mr. Buttigieg have been within striking distance.

New Hampshire Democratic Party Chairman Raymond Buckley said it is “highly unusual to have at this point so many competitive campaigns. Usually we are down to three at the most.”

Write to John McCormick at mccormick.john@wsj.com, Julie Bykowicz at julie.bykowicz@wsj.com and Chad Day at Chad.Day@wsj.com