Preheat the oven to 350 degrees F (175 degrees C). Grease a 4x8-inch loaf pan.

Beat white sugar and butter together in a large bowl with an electric mixer until light and fluffy. Add eggs, one at a time, beating well after each addition. Stir in zest and juice of 1 lemon.

Whisk flour, baking powder, and salt together in a separate bowl; add to butter mixture alternately with milk, mixing until batter just incorporated. Fold in coconut until just evenly combined; pour into prepared loaf pan.

Bake in the preheated oven until a toothpick inserted into the center comes out clean, 60 to 65 minutes. Cool in the pan for 10 minutes. Run a table knife around edges to loosen. Invert carefully onto a wire rack. Let cool completely.

Slowly whisk 2 teaspoons lemon juice into confectioners' sugar in a bowl until thin enough to drizzle; transfer to a resealable plastic bag. Cut a small hole in one corner of the bag; gently squeeze bag and drizzle glaze evenly over loaf.