Melt butter in a 5-quart saucepan over medium heat. Cook and stir onion, celery, and garlic in melted butter until onion is tender, 5 to 7 minutes.

Stir flour into the saucepan until vegetables are evenly coated. Pour in clam juice and water; whisk until smooth. Add potatoes, bay leaf, and thyme; bring liquid to a boil. Reduce the heat to medium-low and simmer until potatoes are tender, about 15 minutes.

Gently stir cod into the soup. Cover the saucepan and cook until cod is flaky, about 8 minutes.

Stir in milk and simmer until soup is hot, 1 to 2 minutes. Season with salt and pepper. Remove bay leaf to serve.