Preheat the oven to 180°C or 350°F. Grease and flour a 9 by 5 inch loaf pan generously.
Sift together the flour, baking powder, and salt into a bowl. In another bowl, whisk together the yogurt, 3/4 cup (150g) granulated sugar, the eggs, 2 teaspoons 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. Do NOT OVER-MIX. Pour the batter into the prepared pan and bake for about 44 to 50 minutes, or until a toothpick placed in the center of the loaf comes out clean.
Whiles the cake is baking, cook the 1/3 cup lemon juice and remaining 1/4cup (50g) granulated sugar in a small pan 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 baking rack.. While the cake is still warm, make small holes in the cake using a toothpick and pour the lemon-sugar mixture over the cake and allow it to soak in.
For the glaze, combine the icing or confectioners' sugar, lemon zest and lemon juice and pour over the cake. Slice and serve. Cake can be stored in the freezer for up to three months and room temperature for three days.