Preheat your oven to 350°F (175°C) and line an 8x8-inch baking pan with parchment paper, leaving an overhang on two opposite sides for easy removal.
In a medium bowl, combine the graham cracker crumbs, 1/2 cup of melted butter, and 1/4 cup of granulated sugar. Stir with a fork until the mixture resembles wet sand and the crumbs are evenly moistened.
Firmly press the crumb mixture into the bottom of the prepared pan to form a compact, even layer. Bake for 10 minutes, then place the pan on a wire rack to cool slightly while you prepare the filling.
In a large bowl, use an electric mixer on medium speed to beat the softened cream cheese and 2/3 cup of granulated sugar until the mixture is completely smooth and creamy, about 2 minutes. Scrape down the sides of the bowl with a spatula.
Beat in the eggs one at a time, followed by the vanilla extract and sour cream, mixing on low speed just until each ingredient is fully incorporated and the filling is uniform. Be careful not to overmix.
Pour the cheesecake filling over the pre-baked crust and spread it into an even layer. Dollop the raspberry jam over the top and use a knife or toothpick to gently swirl it into the filling. Sprinkle the fresh raspberries evenly over the surface.
Bake for 30-35 minutes, or until the edges are set but the center still has a slight jiggle when you gently shake the pan. The top should look dry and not wet.
Allow the bars to cool completely in the pan on a wire rack, then refrigerate for at least 4 hours, or preferably overnight, until thoroughly chilled and firm before slicing into squares.