Bring a pot of salted water to a boil and add the kale and Swiss chard. Cook for 20 minutes. Add the potatoes to the pot and cook until softened, about 20 more minutes. Strain and reserve the vegetables.
Meanwhile, place the oil in a pot over medium heat. Add the bacon and cook until crisp, about 5 to 7 minutes. Remove with a slotted spoon and drain on paper towels. Keep the bacon warm. Add the garlic to the oil and remove from the heat to infuse the oil. Remove the garlic with a slotted spoon and add the strained vegetables to the pot of infused oil. Mash the vegetables until thoroughly combined, but still a bit chunky.
To serve, mound the vegetables on a serving platter and garnish the top with the reserved bacon. Serve warm or room temperature.