Preheat the oven to 350° F/175° C. Grease the loaf pan. Line the bottom with parchment paper, then grease and flour the pan.
In the medium mixing bowl, sift together the flour, baking powder, and salt.
In the large mixing bowl, whisk together the yogurt, sugar, eggs, lemon zest, and vanilla. Slowly whisk the dry ingredients into the wet ingredients. With a rubber spatula, fold the vegetable oil into the batter, making sure it's all incorporated. The oil may not combine at first, but continue until the oil is fully mixed in.
Pour the batter into the prepared pan and bake for about 50 minutes, or until a cake tester placed in the center of the loaf comes out clean.
While the cake bakes, cook the lemon juice and sugar in a small pan over low heat until the sugar dissolves and the mixture is clear. Set aside.
When the cake is done, allow it to cool in the pan for 10 minutes. Carefully place on a wire colling rack over a sheet pan. While the cake is still warm, pour the lemon-sugar mixture over the cake and allow it to soak in, then allow the cake to finish cooling.
In a medium mixing bowl, whisk together the sifted confectioners' sugar and lemon juice. Pour over the cooled cake and allow the glaze to set up.