Remove leaves from collard, mustard, and turnip greens; discard stems. Gently wash leaves in warm water to remove all soil and sand. Partially fill a clean sink with warm water; stir in 2 cups chicken broth and 3 tablespoons salt. Allow greens to soak in chicken broth mixture for 10 minutes.
Scoop greens into a colander and allow to drain; do not rinse. Discard chicken broth mixture down the sink.
Pour remaining 2 cups chicken broth into a large pot; mix in greens.
Stir in vegetable oil, bacon bits, sugar, garlic, salt, and pepper; bring to a boil.
Reduce heat to a simmer and cook, stirring occasionally, until greens are tender, 45 minutes to 1 hour.