Warm the oil in a large skillet over medium heat.
Add onion and cook until softened and translucent, about 4 minutes.
Stir in garlic and ginger; cook until fragrant, about 30 seconds.
Add ground turkey, salt, and pepper. Cook, breaking it up, until browned, 5–6 minutes.
Sprinkle in curry powder, cumin, paprika, and red pepper flakes if using. Toast 30 seconds.
Stir in tomato paste and cook, stirring constantly, for 1 minute.
Pour in diced tomatoes, coconut milk, and broth. Scrape up browned bits from the pan.
Simmer uncovered, stirring occasionally, until slightly thickened, 8–10 minutes.
Fold in peas and spinach; cook until peas are hot and spinach wilts, about 2 minutes.
Remove from heat. Stir in lime juice and cilantro; taste and adjust seasoning.