We study smooth projective threefolds fibered by toric surfaces over the projective line. We show that for certain families of degree 6 del Pezzo and quadric surface bundles the universal torsor corresponding to the generic fiber extends to a smooth model over the base. It respects the action of model for the Neron-Severi torus and induces the Abel-Jacobi map from the space of sections. This corresponds to the map from a set of rational points on the generic fiber to the Galois cohomology group of torsors under the Neron-Severi torus. For the model of the latter we also compute corresponding groups of torsors over the base.