Or give them something to watch while the magic happens. A mesmerising animation, for example. The most important thing is to convey that nothing is broken, and progress is being made.

Show photos of landmarks of the place they are searching flights to.

Maybe do like the old Sims games:

"Upholstering Airline Seatbacks..." "Recalculating Airspeed Velocity..."

Maybe too cheesy, but it would make me laugh.

"Reticulating splines"

What about scrolling through testimonials from satisfied customers?

Have the testimonials fly by in an airplane ;)

This. This method is super proven to help consumer happiness.

Right! Like a chipmunk in a cute scarf or something.