Heat oil in a large pot over medium heat. Sauté onion, bell pepper, garlic, and ginger until soft, about 3-4 minutes.
Add curry paste and cook for 1 minute, stirring constantly until fragrant.
Add carrot, kabocha squash, coconut milk, and water. Bring to a boil, then reduce heat and simmer for 15 minutes.
Add baby corn, bamboo shoots, and broccoli. Continue simmering until all vegetables are tender, about 5-7 minutes.
Remove from heat and stir in Thai basil and cilantro before serving with lime wedges and rice.