Place the water and turkey legs in an 8-quart pot over medium-high heat. Cover, bring to a boil, and allow to simmer for 10 minutes.
In the meantime, remove any large stems from the greens and wash them thoroughly in a sink filled with at least 5 inches of water. Move the leaves around in the water, then let them sit for a few minutes to allow any sand or dirt to fall to the bottom of the sink. Once clean, chop pieces in half. You should have 2 pounds of greens once they are stemmed. (Weigh the greens after stemming, but before washing.)
Once the turkey legs have simmered for 10 minutes, add the greens, salt, and sugar, reduce the heat to low, cover, and allow to simmer gently until the greens are tender, about 45 minutes, stirring the greens every 10 to 15 minutes. Taste and season with additional salt, if desired. Serve immediately.