Heat the oil in a large pot over medium heat and cook the onion until soft, about 5 minutes. Add the garlic and ginger and cook 1 minute.
Stir in curry paste, cumin, and turmeric; cook a full minute until fragrant to deepen flavor.
Add the rinsed lentils and broth. Bring to a boil, then reduce to a simmer, cover, and cook 20 to 25 minutes until the lentils have completely fallen apart.
Stir in the coconut milk and warm through without boiling. Add the spinach and stir until just wilted.
Finish with lime juice and salt; taste and add lime until bright, not heavy.