Preheat the oven to 325 degrees F (165 degrees C). Grease and flour a 9x5-inch loaf pan.

Whisk flour, salt, baking powder, and baking soda in a bowl. Cream sugar and butter in a separate bowl with an electric mixer. Add eggs, lemon juice, lemon zest, lemon extract, and vanilla; beat until incorporated.

Pour the egg mixture into the flour mixture; whisk until smooth. Add oil and mix well. Pour into the prepared loaf pan.

Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 45 minutes. Cool on a wire rack.

While the cake is cooling, mix powdered sugar, milk, lemon zest, and lemon extract in a bowl until smooth. Pour over the cooled loaf. Slice up the loaf when the icing sets.