Preheat the oven to 350°F (175°C). Grease and line a 9x5-inch loaf pan with parchment.
In a bowl whisk together 1½ cups flour, 1 tsp baking powder, ½ tsp baking soda and ¼ tsp salt.
In a large bowl beat ½ cup softened butter with ¾ cup sugar until light and fluffy, about 2–3 minutes.
Add eggs one at a time, mixing well after each addition, then stir in 1 tsp vanilla and the lemon zest.
Stir in 1/3 cup fresh lemon juice to the wet mixture.
Alternate adding the dry ingredients with ½ cup yogurt and ¼ cup milk, beginning and ending with the dry ingredients; mix until just combined.
Scrape the batter into the prepared loaf pan and smooth the top with a spatula.
Bake for 45–55 minutes, checking at 40 minutes; a toothpick inserted should come out clean or with moist crumbs.
Cool in the pan 10 minutes, then transfer to a wire rack to cool completely.
Make the glaze by whisking 1 cup powdered sugar with 2–3 tbsp lemon juice until smooth; pour over the cooled loaf and allow to set before slicing.