Birds don’t have a urinary bladder. Doing away with that is one of the hallmark adaptations of birds for flight, as a urine-filled bladder would weigh them down. They excrete their "urine" as a white paste, resembling toothpaste, mixed with feces, from a single opening called a vent, rather than an anus—although some sources, like the one I quote next, take a little liberty with the terms.
Contrary to another answer already given to this question, one ornithology website speaks of the birds’ “outer anal opening, which is closed by a strong sphincter muscle" (known as the cloaca). From my observations of birds, I feel sure that is correct.
Among other such observations, I’ve often watched the ground-nesting blue-footed boobies in the Galápagos Islands. When they’re sitting on the nest and have to defecate, they stand up, point their tail end away from the sun, and let out a white pasty squirt of mixed feces and urine. As the sun crosses the sky, they face in a different direction hour after hour, like feathered sundials, so the white guano forms a spoke-like array around the center of the nest. I don’t think they could do this if they didn’t have control over an anal (cloacal) sphincter.
I know also that many birds defecate just before they take flight (so do bats) in order to lighten the body a little. In tree-nesting birds (cup nests in branches, stick-nests of hawks and eagles, tree-hole nests of owls and woodpeckers), when the bird has to defecate, it stands up and hangs its tail end over the edge of the nest or outside the rim of the tree hole and lets go, rather than defecating in the nest. When seagulls and terns mob a person or a predator, they hover overhead and defecate profusely on their enemy. These behaviors, too, speak of a sphincter with voluntary control.
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