Let's get some discussion going on this, shall we?
Woodland Park has a number of very impressive bird exhibits (penguins, sea eagles and various breeding cranes being the highlights) but I once read that Taronga has well over 100 bird species and many species that are found nowhere else outside of Oz.
My vote is 2-1 Taronga Zoo
According to this 2019 species list for Woodland Park (credit due to
@Anteaterman), Woodland Park may not be far off in numbers. See below:
Ostrich
Helmeted Guineafowl
Egyptian Goose
Hottentot Teal
White-Faced Whistling Duck
Spur-winged Lapwing
Speckled Mousebird
Golden-Breasted Starling
Taveta Golden Weaver
White-Headed Buffalo Weaver
Maccoa Duck
Snowy-Crowned Robin-Chat
Violet Turaco
Blue-Winged Goose
Chinese Hwamei
Oriental Magpie-Robin
Cabot's Tragopan
Great Blue Turaco
Common Trumpeter
Lady Ross' Turaco
Victoria Crowned Pigeon
Bali Mynah (Off Exhibit)
Palawan Peacock Pheasant (Off Exhibit)
White-Crested Laughing Thrush
Malay Great Argus
Edward's Pheasant
Green Woodhoopoe
Eastern Crested Guineafowl
Vulturine Guineafowl
Blue-Billed Curassow
Bornean Crested Fireback
Blue Magpie
Temminck's Tragopan
Tawny Frogmouth
Nicobar Pigeon
Green-Naped Pheasant Pigeon
Red-Crowned Crane
Swan Goose
Barnacle Goose
Trumpeter Swan
Emperor Goose
Red-Breasted Goose
Smew
White-Headed Duck
Mandarin Duck
Marbled Teal
Northern Shoveler
Pacific Eider
Mandarin Duck
Eurasian Wigeon
Falcated Duck
Green-Winged Teal
Baer's Pochard
Baikal Teal
Bufflehead
Scaly-Sided Merganser
White-Naped Crane
Chilean Flamingo
Puna Teal
Chiloe Wigeon
Southern Screamer
Humboldt Penguin
Sunbittern
Red-Crested Cardinal
Silver-Beaked Tanager
Turquoise Tanager
Blue-Grey Tanager
Red-Crested Finch
Green Aracari
Toco Toucan
Croaking Ground Dove
Paradise Tanager
Red-Capped Cardinal
Golden-Bellied Grosbeak
Yellow-Rumped Cacique
Andean Cock-of-the-Rock
Crested Oropendola
Bananaquit
Rufous-Collared Sparrow
Chaco Chachalaca
Spangled Cotinga
White-Rumped Shama Thrush
Azure-Winged Magpie
Bar-Headed Goose
Demoiselle Crane
Kea
Budgerigar
Cockatiel
Eastern Rosella
Princess Parrot
Kakariki (Red-Fronted Parakeet)
Wonga Pigeon
Laughing Kookaburra
Blue-Faced Honeyeater
Masked Lapwing (No Signage)
Great Gray Owl
Snowy Owl
Steller's Sea Eagle
I tried to cut out any duplicates; if I was successful, that should be
98 bird species for Woodland Park - an impressive number for even a large American zoo. Not only that, but Woodland Park seems to have a large and diverse collection from multiple continents: Africa, Asia, Australia, and the Americas. Anyone can correct me if I have misinterpreted, but aren't Australian zoos somewhat challenged by the import situation when it comes to birds from foreign countries?
Woodland Park also has a number of very choice species: not just Steller's sea eagle and cock-of-the-rock, but also Snowy Owl, Bananaquit, Great Blue Turaco (+2 other species), 3 species of guineafowl, green woodhoopoe, robin-chat, multiple species of magpie, a huge waterfowl collection, 3 species of crane, 2 species of tragopan, 2 species of weavers, 2 species of cardinal, 4 species of tanager, Chinese hwamei, blue-billed curassow... I could go on, but hopefully the point has been made
Here are some exhibits from Woodland Park to give perspective on that category. All have been provided by
@snowleopard, except for the Humboldt penguin photo which came from
@Arizona Docent:
Rainforest Aviary
Tropical Rainforest building exhibit (2014, 4 species)
Australian Aviary
Crane exhibit
One of the Conservation aviaries
Steller's Sea Eagle aviary
Humboldt penguin exhibit (chosen for cool snow backdrop, couldn't find a photo that showed even the majority of the enclosure)
As for conservation, many on this site may be more familiar with Woodland Park's conservation work than its species or exhibits. Birds is no different a story. The zoo works with native raptors in eastern Washington, cranes in the Amur River Basin, Humboldt penguins in Peru, tracking and surveys of Steller's sea eagles, Asian songbird conservation in Bali, and red-breasted goose conservation in Bulgaria.
TL;DR: around 100 species representing many different taxonomic groups and geographic regions, many very good indoor exhibits and outdoor enclosures that mesh well with the zoo's natural biome, and a wide range of bird-related conservation programs. For now, I will be voting
3-0 for Woodland Park. That being said, not much has been posted for Taronga yet; if others step up and make convincing arguments, I will adjust my vote accordingly.