Preheat oven to 350°F. Grease a 9x5-inch loaf tin and line with a sheet of parchment going across the long sides, leaving some overhang to lift the loaf out of the tin when its baked.
Combine the shredded carrots with the lemon juice and a hefty pinch of salt (you can do this in the bowl of the food processor) and allow to sit while you make the batter.
In a large mixing bowl, whisk together the flour, salt, baking soda and baking powder.
In a medium mixing bowl, whisk together the sugars with the turmeric, cinnamon, and ginger, breaking up any lumps. Whisk in the eggs, vanilla, oil and yogurt until smooth. Fold in the carrots and any juices that have accumulated.
Make a well in the dry ingredients. Pour the carrot mixture into the well and stir working in a circular motion from the inside out, slowly bringing the dry into the wet ingredients. Continue folding, scraping the bottoms and sides of the bowl, just until there are no remaining streaks of flour.
Pour the batter into the prepared tin and level it off. With a greased knife, cut a shallow line running down the center of of the batter (this helps to create an even crack — you can also do this by piping a thin line of softened butter down the center, as seen above).
Bake on the middle rack for about 55 to 60 minutes for a 9x5-inch tin, or about 60-65 minutes if using a taller, pullman-style tin. Check the internal temperature of the cake in the very center; it should read around 200-205°F. Alternatively, a wooden skewer placed into the center of the cake should come out clean or with just a few moist crumbs.
Allow the cake cool in the tin for 5 minutes then lift it out of the tin using the parchment sling and allow to cool completely on a wire rack.
Once the cake has cooled completely, make the glaze. Mix together the powdered sugar, lemon juice, and turmeric (start with 1/8 teaspoon and add more if desired). The glaze should be very thick but still relax into itself after 5 seconds or so. If it’s too thick, add a few more drops of lemon juice. Spread evenly over the top of the loaf, encouraging it to drip down the sides. Allow the glaze to set, and serve!