I tracked down all of the major American zoos that have at least 100 bird species and I've typed out a list below. Other than a few exceptions where data was unavailable, this is a fairly comprehensive list and naturally includes behind-the-scenes animals. I obtained this data from the International Zoo Yearbook and San Diego Zoo leads the list of 30 facilities. In fact, between the two 'San Diego Zoo Global' zoological attractions there are a ton of birds!
San Diego Zoo - 330 species
Bronx Zoo - 300
Zoo Miami - 230
San Antonio Zoo - 225
San Diego Zoo Safari Park - 215
Houston Zoo - 205
Omaha's Henry Doorly Zoo - 200
Toledo Zoo - 175
Cincinnati Zoo - 160
Fort Worth Zoo - 160
Denver Zoo - 150
Saint Louis Zoo -150
Sedgwick County Zoo - 150
Wildlife World Zoo & Aquarium - 150
Columbus Zoo - 140
Dallas Zoo - 140
Disney's Animal Kingdom - 140
Busch Gardens Tampa Bay - 130
Zoo Tampa at Lowry Park - 130
Phoenix Zoo - 120
Memphis Zoo - 115
Tulsa Zoo - 115
Woodland Park Zoo - 115
Honolulu Zoo - 110
Birmingham Zoo - 105
Kansas City Zoo - 105
Audubon Zoo - 100
National Zoo - 100
Philadelphia Zoo - 100
Rio Grande Zoo - 100
Notable absentees include Dallas World Aquarium (no data is ever available from that place) and specialist bird zoos such as Tracy Aviary, Sylvan Heights Bird Park and the National Aviary in Pittsburgh.
San Diego Zoo - 330 species
Bronx Zoo - 300
Zoo Miami - 230
San Antonio Zoo - 225
San Diego Zoo Safari Park - 215
Houston Zoo - 205
Omaha's Henry Doorly Zoo - 200
Toledo Zoo - 175
Cincinnati Zoo - 160
Fort Worth Zoo - 160
Denver Zoo - 150
Saint Louis Zoo -150
Sedgwick County Zoo - 150
Wildlife World Zoo & Aquarium - 150
Columbus Zoo - 140
Dallas Zoo - 140
Disney's Animal Kingdom - 140
Busch Gardens Tampa Bay - 130
Zoo Tampa at Lowry Park - 130
Phoenix Zoo - 120
Memphis Zoo - 115
Tulsa Zoo - 115
Woodland Park Zoo - 115
Honolulu Zoo - 110
Birmingham Zoo - 105
Kansas City Zoo - 105
Audubon Zoo - 100
National Zoo - 100
Philadelphia Zoo - 100
Rio Grande Zoo - 100
Notable absentees include Dallas World Aquarium (no data is ever available from that place) and specialist bird zoos such as Tracy Aviary, Sylvan Heights Bird Park and the National Aviary in Pittsburgh.