Preheat the oven to 350 degrees Fahrenheit. Grease a loaf pan that measures 8 1/2 by 4 1/4 by 2 1/2 inches. Line the bottom of the pan with parchment paper.
Combine the flour, baking powder, and salt in a bowl. Mix yogurt, 1 cup sugar, eggs, lemon zest, and vanilla in another bowl.
Add the dry ingredients to the wet ingredients and whisk slowly. Fold the vegetable oil into the batter with a rubber spatula.
In the prepared pan, pour the batter and bake in preheated oven for about 50 minutes, or until a cake tester inserted into the center comes out clean.
Meanwhile, in a small saucepan, dissolve the sugar and 1/3 cup of lemon juice until the mixture is clear. Keep aside.
Allow the cake to cool in the pan for 10 minutes after it has been baked. Place carefully on a sheet pan over a baking rack.
Pour the lemon-sugar mixture over the warm cake and let it soak in. Allow to cool.
Combine confectioners' sugar and lemon juice for the glaze. Pour the glaze over the cake. Enjoy!