Under a fresh cloud of overseas violence, Donald Trump and Hillary Clinton padded their delegate troves on Tuesday with victories in Arizona and attacked each other's worldviews as the 2016 presidential contest turned into a clash of would-be commanders in chief.

While both front-runners scored victories in the night's biggest prize of Arizona, Democratic challenger Bernie Sanders won caucuses in Utah and Idaho and Republican Ted Cruz claimed his party's caucuses in Utah.

The victories kept Clinton and Trump from dominating another election night, but they both maintained a comfortable lead in the race for delegates that decide the presidential nominations.

The Republican primary in Arizona was a winner-take-all affair, with Trump banking 58 delegates to his total.

Cruz countered by capturing all of Utah's 40 delegates, easily meeting a threshold of at least 50 per cent of the caucus vote. Otherwise, the delegates in Utah would have been assigned proportionally.

Ohio Gov. John Kasich, who is trying to position himself as a possible alternative at a brokered convention, was on pace to earn just 10 per cent of Arizona votes, and 17 per cent in Utah.

Overall, Trump has accumulated 739 delegates, Cruz has 465 and Kasich 143. It takes 1,237 delegates to win the GOP nomination.

Sanders correctly predicts 2 wins

While Sanders won two of three states, the delegate math was basically a wash given Arizona's greater size.

Sanders won 57 delegates to Clinton's 51 on Tuesday. Including superdelegates, she leads the Vermont senator 1,681 to 927.

But Sanders was determined to carry on.

"These decisive victories in Idaho and Utah give me confidence that we will continue to win major victories in the coming contests," he said in a statement.

DEM 2016 Sanders

Democratic presidential candidate Senator Bernie Sanders speaks at a rally in San Diego on Tuesday. (Lenny Ignelzi/The Associated Press)

Long lines and high interest marked the races, but the mood shifted overnight as the world grappled with a new wave of bloody attacks in Europe, this time in Brussels.

"What we saw happen today in Brussels, the horrible terrorist attack, reminds us of how high the stakes are," Clinton said at a rally in Seattle, proceeding to take a shot at Republicans Trump and Cruz.

"The last thing we need, my friends, are leaders who incite more fear," she added. "In the face of terror, America doesn't panic, we don't build walls or turn our backs on our allies."

There was a frenzy of activity in Utah as voters lined up to caucus and the state Democratic Party's website crashed due to high traffic. In Arizona, voters waited two hours to cast primary ballots in some cases, while police were called to help with traffic control and at least one polling place ran out of ballots.

Cruz, Trump in Twitter spat over wives

In Arizona, controversial sheriff Joe Arpaio said the Republican Party should heed the wishes of Trump's supporters and not think of denying his nomination at the party's convention this summer in Cleveland.

Arpaio, the six-term official, is known for immigration crackdowns that made him popular among voters, but which were eventually barred by the courts. He told The Associated Press that Trump's victory came as a result of his views on illegal immigration, experience in real estate development and forceful personality that leads supporters to believe he can carry out his promises.

Donald Trump

Donald Trump did not speak at a rally on Tuesday night, but gave an interview to Fox News and engaged in a Twitter spat with main rival Ted Cruz. (Evan Vucci/Associated Press)

Trump's brash tone has turned off some Republican voters in Utah, where preference polls suggesting Cruz would meet the winner-take-all threshold were proven correct.

Cruz seized on Trump's foreign policy inexperience while declaring that the U.S. is at war with the Islamic State group.

"He doesn't have the minimal knowledge one would expect from a staffer at the State Department, much less from the commander in chief," he told reporters. "The stakes are too high for learning on the job."

The debate between the two took a detour late Tuesday night as they engaged in an unusual Twitter exchange about their wives.

The billionaire warned Cruz he would "spill the beans on your wife" after an anti-Trump outside group ran an ad in Utah featuring Trump's wife, Melania, in a photo shoot that ran in GQ magazine more than a decade ago.

Cruz shot back with a tweet of his own, saying in part, "Donald, if you try to attack Heidi, you're more of a coward than I thought."

With files from CBC News