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Southwest Airlines vows to return to its normal schedule after mass cancellations of its flights created chaos for thousands of people.

Operations "with minimal disruptions" would resume Friday, the company said, adding that it was eager to get back to normalcy for the next holiday weekend.

The carrier had cancelled more than 10,000 flights since a winter storm struck the US last week.

It twice apologised, but faced deep criticism from stranded travellers.

On Thursday, Southwest said it was "encouraged by the progress we've made to realign Crew, their schedules, and our fleet" but admitted that "even our deepest apologies - to our Customers, to our Employees, and to all affected through this disruption - only go so far", and promised to improve.

The airline had earlier said sorry to customers, with the company's chief commercial officer pledging on Wednesday to "work day and night" to regain customer trust and its CEO issuing an apology the previous day.

But the low-cost airline - the largest carrier in the US - struggled to return to a normal flight schedule after the fierce weather swept the country, even as other airlines caught up following the travel disruptions.

The 2,360 flights it cancelled on Thursday comprised nearly all of the total US flight cancellations for the day, according to tracking service FlightAware.

The week's worth of disruptions left customers across the US missing their luggage, scrambling for alternative ways to travel or stranded at airports.

Patrick Keane exclaimed with joy upon being reunited was with his lost luggage and said he had not had his belongings in a week.

"I've just been wearing other people's clothes," he told Reuters.

Others took to social media to say they ended up paying for costly new flights or had to find alternative means of transportation.

Southwest has said it will honour "reasonable requests" for reimbursement for meals, hotel and alternate transportation for travellers stranded between 24 December and 2 January, and will offer flexibility for passengers to make changes to their travel online.

Other major airlines - including United and American Airlines - have pledged to place price caps on flights in certain cities where Southwest operates to help some passengers.

The flight chaos drew criticism from the White House, which has promised to hold the company accountable and ensure they provide vouchers for travel costs incurred as a result of the cancellations.

Speaking to "Good Morning America" on ABC News on Wednesday, Transportation Secretary Pete Buttigieg called the flight cancellations a "meltdown" and argued it was "past the point" where the company could blame bad weather for the disruptions.

Unions and analysts say a variety of factors led to the cancellations, including the airline's "point-to-point" system for how it designs its routes, which they say can leave it vulnerable to staffing issues when it comes to delays.