Preheat oven to 350 degrees F (175 degrees C). Grease a 4×8-inch loaf pan.
Beat white sugar and butter in a large bowl until light and fluffy. Add eggs, one at a time, beating well after each addition. Stir juice and zest of one lemon into butter mixture.
Mix flour, baking powder, and salt in a separate bowl; add to butter mixture alternately with milk, mixing until batter is just incorporated. Fold in coconut, mixing just enough to evenly combine. Pour the batter 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 before removing to cool completely on a wire rack.
Gradually stir 2 teaspoons lemon juice into confectioners’ sugar in a bowl until mixture is thin enough to drizzle. Transfer mixture to a resealable plastic bag. Cut a small hole in the corner of the bag. Gently squeeze bag and drizzle mixture evenly over the loaf.
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