In a large mixing bowl, whisk together the flour, sugar, baking powder, and salt until no lumps remain.
Add the yogurt and stir with a wooden spoon or spatula until a shaggy dough forms. Turn out onto a lightly floured surface and knead with floured hands until smooth and elastic, about 1 minute. The dough should be soft and slightly tacky but not sticky. Add a sprinkle of flour if needed.
Divide the dough into 4 equal pieces, shape into balls, and place on a lightly floured surface. Cover with a clean kitchen towel and let rest at room temperature for 15 minutes.
Working with one piece at a time on a lightly floured surface, roll the dough into an oval or round about 1/4 inch thick.
Heat about 1 tablespoon of the olive oil in a large cast-iron or nonstick skillet over medium heat. Add one piece of dough and cook undisturbed until the surface is puffed in spots and the underside has deep golden-brown patches, about 2 minutes. Flip and cook until the second side is browned and the bread is cooked through but still pliable, 1 to 2 minutes more, adjusting the heat as needed.
Transfer to a wire rack or plate and repeat with the remaining dough, adding more olive oil to the pan as needed. Let cool slightly, then cut each flatbread into quarters.