Airlines have found that they can’t sustain the higher levels of flying they had hoped to offer to capitalize on rising demand. Staffing shortfalls, training logjams and constraints at overwhelmed European airports in particular are stifling their resurgence and forcing them into more restraint.

American Airlines Group Inc.,

AAL -7.43%

United Airlines Holdings Inc.

UAL -10.17%

and

Alaska Air Group Inc.

ALK -0.48%

said this week that their revenue during the second quarter was the highest ever for this time of year, as resurgent demand allowed them to charge high enough fares to cover higher costs. Fears of a recession might be mounting, but consumers still want to travel, airline executives said.

“There are two macro demand trends, recession versus continuing Covid recovery, working at cross-purposes,” United Chief Executive

Scott Kirby

said Thursday. “For now at least, the Covid-19 recovery trend is at least canceling out and arguably exceeding the economic headwinds.”

Still, airlines are reining in their schedules for at least the rest of this year—not because they can’t fill their planes, but to avoid costly operational stumbles.

Robert Isom,

American’s chief executive, said: “We’ve pulled out some additional flying, and that’s flying that we would rather do.”

American said domestic leisure travel has surpassed 2019 levels and remains strong. But the airline said its third-quarter flying capacity will be 8% to 10% lower than in the same period in 2019 as the airline pulls back capacity the rest of this year to build additional buffer into its schedule.

The weather in June was challenging, with significant issues on 27 of 30 days that overwhelmed the airline’s efforts to ensure it would be resilient enough to handle challenges, Mr. Isom said. American canceled over 5% of mainline flights that month, according to FlightAware.

“The month of June was really hard on the airline,” Mr. Isom said. “I can’t—nor can anyone else—do anything about 27 out of 30 days of really severe weather in a number of our hubs,” he said.

American posted a profit of $476 million in the second quarter. United reported a $329 million profit on Wednesday, and

Delta Air Lines Inc.

last week reported a $735 million profit.

Airline shares fell Thursday even as the broader market turned higher. United fell 9.7% to $37.65 while American fell nearly 8% to $14 as of midday. Alaska, which also reported earnings Thursday, fell 1.7%.

United and Delta have also said they would cap growth in the coming months to run more reliably after a difficult period in May and June.

Those pullbacks and efforts to avoid delays and cancellations add to the cost pressures airlines are facing. Delta said last week it expects to pay $700 million in overtime and premium pay to help avert disruption, and United executives said Thursday that the airline will be overstaffed while it gives priority to reliability over growth.

“There is weather, and people do call in sick, and sometimes the jet bridge breaks and the power goes out for 20 minutes and stuff happens. And the system just doesn’t have any buffer to deal with that,” Mr. Kirby said.

That could come to a head after Labor Day. Typically, that is when business travelers hit the road again after children go back to school and family vacations slow down. But corporate travel, while increasing, hasn’t returned to prepandemic levels.

Andrew Nocella,

United’s chief commercial officer, said Thursday that the rate of improvement has slowed in recent weeks.

Airlines might have more trouble passing their higher costs along to consumers if demand slows, said

Eric Bernardini,

leader of the aerospace, defense and aviation practice at AlixPartners, a consulting firm. “It’s another ballgame when you are in a recession and people are going to look at what’s left in their wallet,” he said.

This summer is the second in a row in which airlines have struggled to ramp their operations back up to match demand. In 2021, travelers returned before airlines could rebuild their ranks. Without enough slack in the system, summer storms and other routine problems became dayslong snafus with thousands of scrubbed flights.

Airlines had been hopeful that they had tackled those problems and were ready for an even busier summer this year. Signs began to emerge this spring that there would be challenges once again. Carriers including

JetBlue Airways Corp.

,

Spirit Airlines Inc.

and

Southwest Airlines Co.

cut back on flying pre-emptively.

Carriers needed to bring on thousands of workers to replenish their ranks after offering buyouts and early retirement packages to slash costs in 2020.

While their staffing levels are once again nearing prepandemic levels after a monthslong hiring spree, some airlines have said they are finding that is no longer enough as they work through big backlogs of training requirements and adjust to a workforce comprised of less experienced employees to get back up to full force.

Airlines have also said that air-traffic control is short staffed, leading to problems in highly trafficked corridors such as Florida and the New York area. The Federal Aviation Administration has said it is working with airlines to ease some disruptions, including increasing staffing at a key air-traffic control facility near Jacksonville. But the agency says those issues don’t account for the majority of recent flight problems.

Acute staffing shortages at major European hub airports have been a big source of the summer’s chaos. London’s Heathrow Airport last week said it would cap the number of departing passengers and asked airlines to stop selling new tickets from the airport for the summer season. Airlines have had to scramble to reschedule, refund or reroute passengers at a time when flights have been full.

“We found that request to be quite disappointing and frustrating on many levels,” said

Nate Gatten,

American’s chief government affairs officer, adding that the airline doesn’t expect similar caps at U.S. airports but could see additional problems in Europe.

Write to Alison Sider at alison.sider@wsj.com