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Hillary Clinton and Donald J. Trump overwhelmed their rivals in the Arizona primaries on Tuesday, according to The Associated Press, a show of might from two presidential front-runners who are hoping to avoid prolonging the nominating contest and begin training their fire on each other.

Mrs. Clinton trounced Senator Bernie Sanders, who had campaigned hard across Arizona, capturing the biggest delegate prize on a night when Western Democrats were also voting in the Idaho and Utah caucuses.

Speaking to supporters in Seattle, Mrs. Clinton acknowledged her success in Arizona but ignored Mr. Sanders. She used her remarks to address the terrorist assault on Brussels and pivot toward an attack on Mr. Trump and Senator Ted Cruz, the two leading Republican contenders.

“The last thing we need, my friends, are leaders who incite more fear,” Mrs. Clinton said. “In the face of terror, America doesn’t panic. We don’t build walls or turn our backs on our allies. We can’t throw out everything we know about what works and what doesn’t and start torturing.”

She added, “This is a time for America to lead, not cower, and we will lead, and we will defeat terrorism and defend our friends and allies.”

Mr. Trump easily defeated Mr. Cruz in Arizona, taking all 58 of its delegates and adding to his growing lead. But Mr. Cruz was vying to capture more than 50 percent of Utah Republicans in the hopes of claiming all 40 of that state’s delegates and limiting Mr. Trump’s gains.

The victories recorded by Mrs. Clinton and Mr. Trump showcased the strengths that have propelled them to huge advantages in their respective nomination fights. Mrs. Clinton once again demonstrated her loyal following among older and nonwhite Democrats, both significant constituencies in Arizona. And Mr. Trump proved his appeal among immigration hard-liners, who make up a significant bloc of Republicans in the border state.

Mrs. Clinton’s triumph in Arizona, with its 75 delegates, not only extended her lead over Mr. Sanders, it offered a psychological boost as she heads into a stretch of contests in states that are likely to favor Mr. Sanders, like Alaska and Washington, where losses could highlight her lingering vulnerabilities with liberals.

A hoarse Mr. Sanders, speaking to thousands of supporters Tuesday night in San Diego, claim some credit for what he called “record-breaking turnouts.” Ignoring his lopsided loss in Arizona, he noted that he had won 10 contests so far and predicted that he would win “a couple more tonight.”

Republicans hoping to stop Mr. Trump suffered another blow as he carried Arizona, receiving more votes than Mr. Cruz and Gov. John Kasich of Ohio combined. If his opponents fail to defeat him in Wisconsin, where voters go to the polls in two weeks, they are unlikely to stop him from clinching the nomination on the last day of voting in June.

Turnout by voters in all three states was indeed unusually high, with long lines — some snaking for several city blocks — at polling places and caucus sites. It was a sign that, even as the front-runners in both parties move closer to clinching their nominations, enthusiasm for those still in the fight has not been dampened.

The ballot counting in Arizona was delayed as officials extended voting to account for people who were waiting in line, for hours in some cases, when the polls closed at 7 p.m. In Utah, some Democratic caucus sites had to print additional ballots to accommodate the turnout surge.

Tuesday’s Western contests came as Mr. Trump and Mrs. Clinton have both demonstrated strength in a string of recent primaries.

Mr. Trump, who won four of the five contests on March 15, including Florida and Illinois, has built a substantial delegate advantage over Mr. Cruz, and he campaigned aggressively in Arizona in the hopes of capitalizing on his success and reinforcing the perception that his nomination is inevitable.

He drew thousands of supporters last weekend to events near Phoenix and in Tucson, both of which drew impassioned protests.

An important part of what has galvanized his admirers and opponents alike is Mr. Trump’s tough talk on immigration, notably his suggestion that the ranks of Mexican migrants to the United States are filled with rapists and murderers and his boast that as president he would build a border wall and make Mexico pay for it.

His harsh stance on the subject also drove some of Arizona’s most prominent immigration hard-liners, including Sheriff Joe Arpaio of Maricopa County, the state’s largest, to Mr. Trump’s side.

The Democratic contenders sought to exploit the alliance between the sheriff and Mr. Trump, hoping to appeal to white liberals and to Hispanics, who make up around 30 percent of Arizona’s population.

“When I see people like Sheriff Arpaio and others who are treating fellow human beings with such disrespect, such contempt, it just makes my heart sink,” Mrs. Clinton said at a rally on Monday in Phoenix.

Mr. Sanders has been just as outspoken about Sheriff Arpaio, who has drawn criticism for his aggressive tactics in arresting and detaining illegal immigrants. “It’s easy for bullies like Sheriff Joe Arpaio to pick on people who have no power,” Mr. Sanders said to supporters on Monday in Flagstaff.

Arizona was the most heavily contested of the three states voting on Tuesday in the Democratic race, which has seen Mrs. Clinton open a nearly insurmountable lead after sweeping all five states that voted on March 15.

Only registered Democrats were allowed to vote in Arizona, potentially handicapping Mr. Sanders, who typically overwhelms Mrs. Clinton among independents.

Utah and Idaho, by contrast, were voting through caucuses, not primaries, and have largely white populations — two frequent indications of success for Mr. Sanders.

But with Democrats allocating their delegates on a proportional basis, Mr. Sanders’s share of the combined 64 delegates offered by Utah and Idaho would do little to dent Mrs. Clinton’s lead of 319 pledged delegates heading into Tuesday’s votes.

Arizona was also the biggest prize on Tuesday for Republicans, with the top vote-getter receiving all 58 of its delegates. Mr. Trump led in the polls there and appeared to enjoy an advantage thanks to the state’s expansive early-voting laws: Republicans were eligible to begin casting ballots, in person or through the mail, 26 days before the primary — a period in which Mr. Trump racked up a series of victories. And by the time polls opened Tuesday morning, more than half of Arizona Republicans had cast their ballots.

Though Mr. Cruz campaigned and ran television ads in Arizona, both he and Mr. Kasich were far more focused on Utah’s 40 delegates.

Like Mr. Sanders, Mr. Cruz has enjoyed an advantage in states that hold caucuses, where his on-the-ground organizational efforts can be particularly effective with the doctrinaire conservatives and Republican activists who tend to dominate those party-building events.

In Utah, Mr. Cruz won the support of Mitt Romney, the 2012 Republican nominee, an important assist in the country’s most heavily Mormon state. Mr. Romney recorded get-out-the-vote calls for Mr. Cruz on Monday.

Mr. Kasich, who had focused most of his efforts in the Midwest and Northeast, surprised some anti-Trump Republicans by competing aggressively in Utah because of the possibility that his success there could threaten efforts to slow Mr. Trump’s march to the nomination.

Mr. Trump, who has struggled in some states with large Mormon populations, mounted only a perfunctory campaign in Utah. And he seemed to write off the state on Friday, at a rally in Salt Lake City — the spiritual home of the Latter-day Saints — when he pilloried Mr. Romney and questioned his faith, saying, “Are you sure he’s a Mormon?”