In a large pot or Dutch oven, heat the vegetable oil over medium heat. Gradually whisk in the flour to make a roux, stirring constantly until it is a deep brown color, about 15-20 minutes. Be careful not to burn it.
Add the chopped onion, green bell pepper, and celery to the roux. Cook, stirring occasionally, until the vegetables are soft, about 5 minutes.
Stir in the minced garlic and cook for an additional minute until fragrant.
Gradually add the chicken broth while stirring to avoid lumps. Then add the diced tomatoes with their juice, okra, Cajun seasoning, thyme, bay leaf, salt, and pepper. Bring the mixture to a boil.
Reduce the heat to low and let it simmer for 20 minutes, allowing the flavors to meld together.
Finally, add the shrimp and cook for another 5 minutes until they turn pink and are cooked through.
Remove the bay leaf before serving. Serve the gumbo over cooked white rice and garnish with chopped green onions.