Preheat oven to 325ºF (163ºC). Line the bottom of an 8 X 4-inch loaf pan with parchment paper. Grease and flour the sides.
In a medium bowl, whisk together the flour, baking powder, salt and baking soda.
In a small bowl, whisk together the milk and sour cream.
In a large bowl, beat the sugar, oil, eggs, lemon zest and strawberry extract until smooth.
Stir in the flour mixture in two additions, alternating with the milk mixture. Stir just until combined. Stir in food color a little at time until your desired shade is achieved.
Transfer the batter to the prepared baking pan. Bake until a toothpick inserted into center of the loaf comes out clean, about 45-50 minutes. Meanwhile, prepare lemon syrup.
Let the cake cool in the pan for 10 minutes, then remove from pan and transfer to a cooling rack. Poke holes all over the top of cake with a skewer.
Brush the warm cake generously with the lemon syrup. Let the cake cool completely, then top with icing.
In a small saucepan, combine the juice of one lemon with enough water to equal 1/4 cup. Add sugar and stir over medium-low heat until sugar is dissolved. Remove from heat.
In a medium bowl, beat the butter, powdered sugar, 1 tablespoon lemon juice, the strawberry extract and a pinch of salt, until smooth. Add more lemon juice until your desired consistency is reached. Mix in gel food color.