Heat the oil (or water/broth if you are preparing this without oil) in a wide saucepan. Add the chopped onion and saute on high heat for about 5 minutes until onions are soft.
Add the coconut milk, curry powder, turmeric, salt and cubed sweet potatoes to the pan. Reduce heat to medium and simmer, covered, for 15 minutes.
Add the broccoli florets, chopped carrots, sliced red bell pepper and garbanzo beans. Simmer, covered, for an additional 10 minutes, stirring in the chopped kale for the last three minutes. Pierce a sweet potato cube with a fork to make sure it is soft and thoroughly cooked. Simmer a little longer if neccessary.