Heat olive oil in a large pot over medium-high heat. Cook and stir bacon in hot oil until crisp, 7 to 10 minutes. Remove bacon from pot, crumble, and return to the pot. Add onion to the bacon; cook and stir until the onion is tender, about 5 minutes. Stir garlic into the bacon mixture; cook and stir until fragrant, about 2 minutes. Add collard greens, turnip greens, and mustard greens; cook and stir until the greens start to wilt, 2 to 3 minutes.
Pour chicken broth into the pot; season with salt, black pepper, and red pepper flakes. Reduce heat to low, cover pot with a lid, and cook at a simmer until the greens are tender, about 45 minutes.