Place the oil in a large saucepan over medium heat and add the onions. Sauté 4 minutes, stirring occasionally. Add the sweet potatoes, garlic, jalapeños, salt, coriander, and turmeric. Sauté 2 minutes longer and add the water and coconut milk.
Increase the heat to medium-high and bring to a boil. Reduce the heat to low, cover, and simmer for 15 minutes. With an immersion blender, puree the soup until smooth.
Stir in the turnip greens and continue to simmer uncovered for 10 minutes. If the soup is too thick, add a bit more water. Divide the soup into serving bowls and top with a squeeze of the lime and a garnish of cilantro. Serve immediately.