Combine turkey broth, lemongrass, ginger, cilantro stems, lime leaves, and red pepper flakes in a pot over medium-high heat. Bring to a simmer, reduce heat to medium-low, and simmer until flavors blend, 20 to 30 minutes.
Stain broth mixture through a fine-mesh sieve, discard vegetables, and return broth to the pot.
Stir turkey, mushrooms, fish sauce, and palm sugar into broth; simmer until mushrooms are tender and turkey absorbs broth flavor, about 15 minutes.
Pour coconut milk, lime juice, green onions, and cilantro leaves into broth. Bring soup to a simmer and drizzle in chili oil.
Notes
You can use white sugar in place of palm sugar, if desired. You can use young lemon leaves in place of the lime leaves, if desired