We prove the validity over R of a CDGA model of configuration spaces for simply connected manifolds of dimension at least 4, answering a conjecture of Lambrechts–Stanley. We get as a result that the real homotopy type of such configuration spaces only depends on a Poincaré duality model of the manifold. We moreover prove that our model is compatible with the action of the Fulton–MacPherson operad when the manifold is framed, by relying on Kontsevich's proof of the formality of the little disks operads. We use this more precise result to get a complex computing factorization homology of framed manifolds.