On a June 30 flight on American Airlines from Dallas to Newark, New Jersey, Joy Gonzalez, an aviation engineer based in Seattle, found herself seated at a window with two older passengers beside her in the middle and aisle seats. In order to gain more social distance, she and the aisle passenger both moved to seats behind them where two rows were empty. But before takeoff, a flight attendant ordered them back to their assigned seats, telling them they had not paid for those exit row seats, which are more expensive.

A second flight attendant listened to Gonzalez’s request, consulted with the other attendants and gave her two options: Take your assigned seat or return to the gate and pay for the exit row. As the flight was on the verge of departing, she sat down.

“The irony of then hearing on the public address system, ‘Your health and safety is our top priority,’” said Gonzalez, who posted photos of the full and empty rows on Instagram. “Behind me, seats went empty and wasted while I was squished and touching someone.”

After the coronavirus pandemic hit, airlines vowed to bring social distancing to the air — even if it wasn’t the full 6 feet recommended by the Centers for Disease Control and Prevention — by reducing capacity and blocking many middle seats.

But freeing up that kind of space is plainly at odds with the airlines’ profit motive, and passengers are finding they may be confined to a cramped seat if they don’t pay for a premium one, though American denies this is their policy.

Ross Feinstein, a spokesman for the airline, wrote in an email that the restriction “appears to be in error, as we are permitting customers to move within the main cabin, including Main Cabin Extra seats,” which include exit rows.

Since April, American Airlines has capped capacity at 85%. As of July 1, according to new guidelines, it began filling its planes.

The exit row issue seems to be rooted in an unsympathetic flight crew, more evidence of the stress on the aviation system that emerged over the holiday weekend in videos and photos of packed flights.

They included a tweet from Sen. Jeff Merkley, D-Ore., with a picture of him on a busy American Airlines flight on July 2, asking: “@AmericanAir how many Americans will die bc you fill middle seats, w/ your customers shoulder to shoulder, hour after hour. This is incredibly irresponsible. People eat &drink on planes &must take off masks to do so. No way you aren’t facilitating spread of COVID infections.”

Capacity flights now have the green light from American and United. Both have announced they will inform travelers when their flights are reaching capacity and allow passengers to rebook on less crowded flights if available without penalty.

In contrast, Southwest Airlines is blocking middle seats through at least Sept. 30. JetBlue is doing the same through July 31. Delta Air Lines has reduced capacity to 60% in its main cabin by blocking middle seats or, in a two-by-two seat row, aisle seats, through Sept. 30.

“These policies are aimed at ensuring extra space for every customer on every flight, regardless of where on the plane a customer may be seated,” wrote Drake Castañeda, a spokesman for Delta, in an email.

“Airlines are between a rock and a hard place,” said Joseph P. Schwieterman, a transportation expert and professor in the School of Public Service at DePaul University in Chicago. “They can’t cover their costs unless their planes are at least three-quarters full, but, for many passengers, the prospect of flying elbow-to-elbow with strangers is a forbidding prospect. There is no obvious way to reconcile this contradiction.”

The problem will only grow as demand for travel grows. On July 2, the Transportation Security Administration reached a high of 764,761 travelers passing through its airport checkpoints, the highest figure since March 18 but still well below the more than 2 million travelers processed the same day a year ago.

Kristi Brooks, who works in financial services in Dallas and has high-level Platinum status on American, recently flew Delta on a business trip to Alaska. Through several connecting flights, she never had another passenger sit next to her and received hand sanitizer and hand wipes on each leg of the trip.

Last week, she flew American to Minneapolis and found the empty seat between her and her boyfriend, which, she said, had been blocked when she booked the tickets in May, was eventually filled. Now she’s considering returning home on Delta.

“People should know there are choices and they should look at them,” she said.