Preheat the oven to 350 degrees F (175 degrees C). Lightly grease a 9x13-inch baking pan with oil.
Remove phyllo sheets from the package. Tear each sheet roughly into shreds and pile up in the baking pan. Let shredded phyllo dry out a little while you prepare the remaining ingredients.
Make the syrup: Combine sugar, water, and zest of 2 lemons in a saucepan over medium-high heat. Bring to a boil and simmer for 8 minutes. Remove from heat and stir in lemon juice from all 4 lemons. Allow syrup to cool while you prepare the cake.
Combine yogurt, oil, sugar, eggs, vanilla extract, baking powder, and the remaining lemon zest in a blender or food processor. Blend together on high speed until frothy, about 2 minutes. Pour lemon-egg mixture over the shredded phyllo in the baking pan. Stir everything together gently until egg mixture is evenly distributed, ensuring there are no dry spots of phyllo.
Bake in the preheated oven until the top is golden and the filling set, about 45 minutes.
Remove lemonopita from the oven and immediately ladle the cooled syrup over the hot cake. Set aside until most of the syrup has soaked in, at least 1 hour. Slice into squares and serve.