Republican Donald Trump and Democrat Hillary Clinton secured major victories in the New York primaries Tuesday night, giving a jolt of life to the front-runners' campaigns.

After a string of losses for both candidates, the New York primaries re-established their dominance in their respective races. New York also serves as a springboard for both Clinton and Trump as they head into a series of primaries where they should continue to perform well.

While the path ahead looks a little easier for both of them, Tuesday's results underscored that neither Trump nor Clinton can afford to take the rest of the primary season for granted. On the Republican side, Trump will still have to contend with two opponents attempting to deny him the 1,237 delegates needed to win the nomination outright. Meanwhile, since Democratic delegates are awarded proportionally, Clinton's New York victory won't push Bernie Sanders out of the race.

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In their victory speeches Tuesday night, both candidates sounded like strident winners but readily acknowledged the fight is not over.

"We don't have much of a race anymore," Trump bragged from Trump Tower in Manhattan, noting that Ted Cruz has been "just about mathematically eliminated" from winning 1,237 delegates.

At the same time, Trump gave a stern warning to those in the Republican establishment who may try to block his path to 1,237. "It's really nice to win the delegates with the votes," he said. "Nobody should be given delegates, which is a ticket to victory but not a fair ticket."

Clinton, meanwhile, told her supporters, "The race for the Democratic nomination is in the home stretch, and victory is in sight." Making a direct appeal to the throngs of Sanders supporters in upcoming primary states like Pennsylvania and California, Clinton said, "I believe there is much more that unites us than divides us."

New York rallies behind Trump but sees a divided GOP

With 93 percent of precincts reporting in New York Tuesday night, Trump had won a solid 60 percent of the vote. With this overwhelming victory, Trump will take home nearly all of New York's 95 delegates.

As it has been in other states, Trump's base of support spanned a wide range of demographic groups. CBS News exit polling shows he won the support of both men and women, college graduates and those without a college degree. Some of Trump's strongest support came from voters who were looking for a candidate who would bring needed change and those who were looking for someone who would tell it like it is. He also won the backing of eight in 10 voters who want the next president to be from outside of politics.

Meanwhile, a recent new CBS News Battleground Tracker poll shows that New York could be the start of Trump's newest winning streak. In Pennsylvania, which votes on April 26, Trump leads Ted Cruz, 46 percent to 26 percent. John Kasich, who grew up in the Keystone State, trails with 23 percent. The poll also found Trump leading in California, which votes on June 7, with 49 percent support.

"We expect we're going to have an amazing number of weeks" ahead, Trump said Tuesday night.

It's already impossible for Kasich to reach 1,237 delegates. And as Trump said Tuesday, it could be impossible for Cruz to reach that number by next Tuesday.

That, however, isn't stopping Kasich or Cruz, who are both interested in securing delegate support through a brokered convention. Voters in New York expressed concern with the process. Around six out of 10 Republican voters said the primary process has mostly divided the Republican party. Just 36 percent said it has energized the party.

Already in Philadelphia on Tuesday evening, Cruz declared 2016 "the year of the outsider," following up with the assertion, "I am an outsider." Lowman Henry, the Cruz campaign's Pennsylvania state chairman, told reporters Tuesday night that the senator has strong prospects in the state, both for its 14 at-large delegates and the delegates awarded at the congressional district level.

Kasich, meanwhile, was in Maryland on Tuesday evening. His chief strategist John Weaver wrote in a memo, "It's now or never to stop Trump and save the Republican Party."

The Trump campaign is taking the challenge seriously. Just this week, the campaign underwent an extensive reorganization, all in an effort to build up Trump's delegate support for the remainder of the primary campaign -- and during the GOP convention, if it comes to that. It's "evolving," Trump said Tuesday night of his changing team.

Sanders' path narrows, but he marches on

For Clinton, Tuesday night's home-state win was yet another reminder of how narrow the path forward is for Sanders. While it's not mathematically impossible at this point for him to win the nomination, Tuesday night's results are a blow to a campaign that earlier this month was suggesting it could potentially win New York state's primary outright.

Clinton's victory was in line with or slightly above what public polling in the state predicted, all of which found her leading by between 10 and 15 points in the final days before the primary. With 94 percent of precincts reporting Tuesday night, Clinton led Sanders by 16 points, 58 percent to 42 percent.

"Today we took Secretary Clinton on in her home state of New York, and we lost," Sanders told reporters at a brief press availability upon landing in Burlington, Vermont.

Sanders looked ahead on the map, saying the race next turns to five East Coast states -- Pennsylvania, Connecticut, Rhode Island, Delaware and Maryland -- on April 26. "We look forward to winning a number of those states," he said.

The Vermont senator added that his campaign has come a "very, very long way" since he announced his candidacy last spring, reiterating that there is still a "path" to the Democratic nomination for him.

Meanwhile, almost simultaneously in New York City, Clinton was claiming victory beyond just New York, proclaiming: "There's no place like home."

In a clear nod to Sanders' past comments about her delegate lead coming largely from the Deep South, Clinton stressed that her campaign has "won in every region of the country." "From the north to the south to the east to the west," she continued, "but this one's personal!"

As in past contests, exit polling data shows that Clinton's victory was powered by strong margins among minorities, older voters and women. Among African Americans, who made up almost a quarter of the New York primary electorate on Tuesday, Clinton led 75 percent to 25 percent; among Latinos, who were 14 percent of the electorate, she led 63 percent to 37 percent. Clinton led among voters over 45 by a 30-point margin, 65 percent to 35 percent; among those over 65 years old, she won by an even bigger 44-point margin (72 percent to 28 percent).

Meanwhile, Sanders continued to do well among white voters and younger voters, though by smaller margins here than in other recent states on the map. Among white voters, Sanders had a slight edge, 51 percent to 49 percent. And among 18- to 24-year-olds, he led by a massive margin: 82 percent of the demographic chose Sanders, compared with 18 percent for Clinton. He won 18- to 44-year-old voters overall by a 14-point margin (57 percent to 43 percent).

Despite the fact that Sanders outspent Clinton on New York TV and radio by almost $3 million since April 5, he and his campaign began lowering expectations before voters even went to the polls. On Sunday, Sanders said on CBS' "Face the Nation" that the closed primary system in New York would make it difficult for him to win there.

"Even here in New York state you have a voting system which makes it impossible for independents to participate in the Democratic primary, that makes it impossible for people to register on the day of the election which many states do, which is going to result in a lower voter turnout than I would like to see," he said.