Republican presidential candidate Donald Trump rolled to big wins in Michigan and Mississippi on Tuesday, brushing off a week of withering attacks from the party's establishment to solidify his front-runner status as four states voted in nominating contests. (Reuters)

Donald Trump won convincing victories Tuesday in the Michigan and Mississippi primaries, as well as the caucuses in Hawaii, suggesting that the intensified GOP establishment assault on Trump’s character and record had not yet wounded the front-runner for the Republican presidential nomination.

In Michigan and Mississippi, Trump galvanized huge populations of white working-class voters with his populist economic pitch, nativist rhetoric and outsider appeal to win by double-digit margins, further solidifying the billionaire mogul’s lead in the rollicking nomination battle.

Sen. Ted Cruz (R-Tex.) came in second in the Hawaii caucuses. But he won Idaho decisively, followed by Trump in second and trailed by Sen. Marco Rubio (Fla.) and Ohio Gov. John Kasich.

But the night’s marquee event was in Michigan, where late returns showed Kasich and Cruz still competing for second place behind Trump. In Mississippi, Cruz ran a distant second. Finishing far behind in both states was Rubio, who was on track to record some of his poorest results of the season and was in danger of not qualifying for a single delegate in either state, nor in Idaho.

Tuesday’s contests come at a critical juncture for the Republican Party. The runaway front-runner only a couple of weeks ago, Trump was forced onto the defensive over the past week by his own missteps and by a barrage of savage attacks from his rivals and opposing super PACs.

But Trump prevailed — and sought to seize the mantle of the party’s presumptive nominee as he claimed victory Tuesday night.

Speaking from his golf club in Jupiter, Fla., Trump delivered an impassioned and colorful defense of his business credentials, his candidacy and his personal brand itself. He vowed to work to reelect fellow Republicans up and down the ballot this fall and argued that his campaign was the only one truly expanding the GOP coalition.

“The turnout has been just massive for every week,” Trump said. “We will take many, many people away from the Democrats.”

“What we’re going to do is beat Hillary Clinton — and we’re going to beat her badly,” he added, referring to the leading Democratic candidate.

With a starkly different fate on Tuesday night was Rubio, who registered embarrassingly low vote totals in Michigan and Mississippi. Late returns showed him running in last place in both states, although he was hopeful of doing better in Idaho and Hawaii, both states where his campaign had made investments.

Rubio, who spent Tuesday campaigning in Florida, where he is under intense pressure to win, sought to brush aside Tuesday night’s results as the returns began rolling in.

“I believe with all my heart that the winner of the Florida primary next Tuesday will be the nominee of the Republican Party,” Rubio told a crowd in Ponte Vedra Beach. He then directly confronted Trump: “It’s not enough to stand up here and say you’re going to make America great again. You deserve to know how.”

Rubio has struggled to recover from a string of poor finishes in recent contests and has been an uneven performer in the two weeks since he went on the offensive against Trump.

In his victory remarks, Trump mocked Rubio for the attacks.

“He became hostile a couple of weeks ago, and it didn’t work,” Trump said. “Hostility works for some people but not for everybody. He would’ve been better off had he kept the original pitter-patter going.”

The scene at Trump’s victory party was surreal, with members of the Trump National Golf Club Jupiter dressed in cocktail attire sipping wine and nibbling from charcuterie boards and fresh fruit.

“What happened to Marco Rubio!? Aww, poor little Marco!” one attendee said after Michigan was called, borrowing one of Trump’s campaign-trail taunts.

Displayed near the candidate’s podium were bottles of Trump-branded wine and Trump-branded water, as well as piles of raw, unpackaged steaks he said were “Trump Steaks,” to push back against detractors who criticized him over those products.

In Michigan, the night’s marquee contest, Kasich was poised to register a relative surprise. The Midwesterner has been largely counted out of the national race, but Kasich campaigned harder across Michigan than any other candidate, holding upbeat town hall meetings throughout the state.

Late returns showed Kasich locked in a close race for second with Cruz, with each receiving about a quarter of the vote. The Ohioan was banking on a strong finish in Michigan to give him a needed jolt heading into his must-win home-state primary next Tuesday.

Addressing supporters Tuesday night in Columbus, Kasich projected victory there in a week.

“Think about where we started,” Kasich said. “In the contest going forward, the three of us that remain — we are in a virtual dead heat.” He was referring to Trump, Cruz and himself — writing off Rubio, whom Kasich’s campaign now sees as a spoiler.

At stake Tuesday were 150 convention delegates, which were to be awarded proportionally based on candidates’ performances by congressional district in each of the four states. Each state has thresholds for receiving delegates; in Michigan, for example, candidates must finish with 15 percent of the vote or better to qualify for delegates.

For Trump, Michigan represented the first test of his electoral strength in the Rust Belt. His populist pitches on trade, economic development and immigration resonated deeply with the working-class voters who flocked to the polls in huge numbers.

Michigan is the kind of Democratic-leaning state — Pennsylvania is another — that Trump and his advisers have argued he could make competitive in a general election.

Trump faced another test in Mississippi, a heavily Republican Bible Belt state where he had long been favored because of his anti-immigration, nativist rhetoric. He held a massive, raucous rally on Monday evening in Madison, Miss.

In both states, early network exit polling reported by CNN showed vast majorities of Republican primary voters were angry or dissatisfied with the federal government.

That data showed that Mississippi primary voters divided sharply along ideological lines between Trump and Cruz, with 46 percent identifying as “very conservative,” the most of any contest this year. Strong conservatives have been Cruz’s best constituency this year, and he led Trump by roughly 10 percentage points in the preliminary data. But Trump led by at least 20 points among Republicans who identify as somewhat conservative or moderate.

Fully 85 percent of the voters in Mississippi’s Republican primary said they were evangelical Christians, the exit polling shows. Cruz has focused on appealing to evangelicals with a socially conservative message, but in Mississippi as elsewhere, Trump appears to have blocked Cruz from gaining an edge. The early data found Trump with a small edge among evangelical Christians and a 2-to-1 lead among non-evangelicals.

Late returns showed Trump winning roughly half of the vote in Mississippi, similar to the landslides he won in Alabama, Georgia and Tennessee last Tuesday. The fact that Trump’s Mississippi margin mirrored his double-digit wins in those states — as opposed to his much narrower, four-point win over Cruz in Louisiana on Saturday — suggested that Trump’s popularity had not slipped among conservatives despite the heavy attacks on him.

Some recent polls nationally and in key states have contained warning signs for Trump, indicating that his refusal to immediately disavow former Ku Klux Klan leader David Duke, his debate-stage quip about his manhood, and fresh attacks on his business dealings and character — or a combination of all three — were taking their toll.

Trump is counting on his big wins in Tuesday’s contests, followed by a strong performance in Thursday night’s debate in Miami, to put himself back in full control of the nominating contest before next Tuesday’s primaries in Florida, Illinois, Missouri, North Carolina and Ohio, which could be determinative.

But the GOP establishment has been trying to keep Trump on his heels. Mitt Romney, the party’s 2012 presidential nominee, recorded phone calls sent to Republicans in Michigan and other states voting Tuesday on behalf of Rubio and Kasich. Romney has not endorsed a candidate, but he has become a fierce Trump critic, and in the calls he urged Republicans to vote against Trump.

“I believe these are critical times that demand a serious, thoughtful commander in chief,” Romney says in the calls. “If we Republicans were to choose Donald Trump as our nominee, I believe that the prospects for a safe and prosperous future would be greatly diminished — and I’m convinced Donald Trump would lose to Hillary Clinton. So please vote tomorrow for a candidate who can defeat Hillary Clinton and who can make us proud.”

Tuesday’s biggest prize was Michigan, which awards 59 of the 150 delegates. Although polls showed Trump with a substantial lead, Cruz made a hastily scheduled stopover in Grand Rapids late Monday, hoping to mobilize conservative voters there.

Michigan has relatively few evangelical voters and is hardly tailor-made for Cruz, though there are strong social conservative and libertarian strains in the Republican base. Cruz saw an opportunity to capi­tal­ize on his gains in last weekend’s contests and take advantage of Rubio’s struggles to finish a strong second.

Contending with him for that position was Kasich, who was on the rise in recent days while approaching the Michigan primary like a governor’s race. He campaigned in every corner — including the remote Upper Peninsula — and racked up a bushel of endorsements from local officials.

The exit polls showed more than 6 in 10 Michigan voters made up their minds well before Tuesday, and Trump won them by a nearly 2-to-1 margin. But Kasich was the chief beneficiary of voters who decided in the final week, winning them with 43 percent compared with 25 percent for Cruz and 18 percent for Trump.

Ed O’Keefe in Miami; Jose A. DelReal in Jupiter, Fla.; David Weigel in Columbus, Ohio; and Scott Clement and Juliet Eilperin in Washington contributed to this report.