Preheat oven to 350°F (175°C) and line a 9x13 inch baking pan with parchment paper.
Combine softened butter, 0.5 cup sugar, 2 cups flour, and salt in a bowl. Mix until a crumbly dough forms.
Press the dough firmly into the bottom of the prepared pan in an even layer.
Bake the crust for 20 minutes or until the edges are very lightly golden brown.
While the crust bakes, whisk together 1.5 cups sugar and 0.25 cup flour in a large bowl.
Add the eggs, lemon juice, lemon zest, and strained strawberry puree to the sugar mixture; whisk until fully combined and smooth.
Pour the strawberry-lemon mixture over the hot crust immediately after removing the crust from the oven.
Bake for an additional 25 minutes or until the filling is set and no longer jiggles in the center.
Remove from oven and let cool completely at room temperature for at least 1 hour.
Refrigerate for 2 hours to ensure clean slicing. Dust with powdered sugar and cut into 16 bars before serving.