Place bacon in a large Dutch oven and cook over medium-high heat, turning occasionally, until evenly browned, about 10 minutes. Remove the bacon slices to a plate lined with paper towels, reserving some of the drippings in the Dutch oven.
Stir onion in reserved drippings; cook and stir until browned, about 10 minutes. Stir garlic into the onion; cook until softened, 1 to 2 minutes. Pour broth into the Dutch oven; add green beans. Chop bacon and add to the Dutch oven.
Bring the broth to a boil, reduce heat to low, and place a cover on the Dutch oven. Simmer the green bean mixture until the beans are tender, about 1 hour. Add potatoes; season with salt and pepper. Continue cooking until the potatoes are tender, about 30 minutes more.