Heat the oven to 350°F. Butter a 9-inch square baking pan and line it with parchment paper, leaving a 2-inch overhang on two sides. In a large bowl, combine the flour, brown sugar, baking powder, and salt. Add the butter and beat with an electric mixture until the mixture is the texture of coarse sand. Sprinkle 2/3 of the mixture into the pan and press into an even layer. Bake until golden brown, 24 to 26 minutes. Add the oats to the remaining crumb mixture.
Prepare the jam: In a medium saucepan, combine the strawberries, rhubarb, and granulated sugar. Cook over medium heat, stirring occasionally, until the mixture is bubbling fruit has broken down, 7 to 8 minutes. In a small bowl, combine 3 tablespoons of the cooked fruit mixture with the cornstarch and stir until smooth. Add the cornstarch mixture back to the jam and simmer until thickened, about 3-4 minutes. Let cool.
Spread cooled jam evenly over the prepared crust and top with remaining crumb mixture so that it forms large clumps. Return to oven and bake until golden brown, 28 to 32 minutes. Transfer to a rack to cool completely. Using the parchment, left the cooled bars from the pan and cut into squares to serve.