Make room for the crest and tail
A cockatiel's carrier has to fit the bird's full outline, crest up and tail stretched behind, which rules out many carriers sized for smaller parrots. When a cockatiel's long tail presses against a closed door for a whole journey, the feathers bend and fray and the bird keeps shifting to escape the pressure, arriving at the clinic restless and ruffled. Choose a carrier or small travel cage long enough that the cockatiel can sit on its perch with the tail clearing the back wall, and tall enough that a raised crest does not brush the roof when the bird startles.
Hard sides and spacing that keeps a head in
A rigid plastic carrier or a well-made travel cage suits a cockatiel better than a soft fabric carrier, whose walls flex inward under a seatbelt and whose mesh a determined beak can work at during a long wait. If the carrier has bars or a barred door, the spacing must be narrow enough that the cockatiel cannot push its head through and wedge itself, yet the door itself should open wide, because cockatiels back away from hands and a wide front lets a nurse encourage the bird out rather than reach in and grab.
One low perch, and the night-fright lesson
Set one perch low and secure in the cockatiel's carrier, fixed so it cannot rotate or slide when the car brakes. Cockatiels are prone to sudden panic flapping, the same reflex behind night frights at home, and in a carrier that flurry is the main injury risk: a low perch shortens any fall, and snug carrier dimensions stop the bird building up speed across the box. Line the floor with paper towel for grip and so droppings stay visible for the vet, and skip toys entirely, since a swinging toy during a panic can strike the bird.
Calm the journey, not just contain it
A light cover over the top and sides of a cockatiel's carrier makes a marked difference, because this species watches everything and startles at movement overhead. Leave the front partly open for air and so the bird can see you if that settles it, and talk at your normal volume rather than whispering, which many owners find makes a cockatiel more suspicious, not less. Place the carrier on the seat, belted or wedged so it cannot tip, and keep the car at a comfortable temperature; a cockatiel that overheats in a closed carrier will pant with its beak open and wings held out.
Arrive with the bird's story ready
The carrier solves the physical trip; the paperwork is the other half of a good vet visit. Note when you last saw the cockatiel eat normally, how its droppings have changed, and anything new in its diet or environment, and bring a fresh sample of the food it eats if the visit concerns weight or digestion. Keeping your cockatiel's records together in a KinsLe Pet Passport, one portable packet ready for vet visits and handoffs, means those details travel with the bird instead of being remembered in the waiting room.