Epitaxial $Ba_{0.5}Sr_{0.5}TiO_3$thin films of two different thickness (~250 $AA$ and ~1340 $AA$) on MgO(001) prepared by a pulsed laser deposition method were studied by synchroton x-ray scattering measurements. The film initially grew on MgO(001) with a cube-on-cube relationship, maintaining it during further growth. As the film grew, the surface of the film became rough significantly, but the interface between the film and the substrate seemed to have changed little. In the early stage, the film was highly strained in a tetragonal structure with the longer axis parallel to the surface normal direction. As the growth proceeded further, it was mostly relaxed to a cubic structure with the lattice parameter of the bulk value and the mosaic distribution improved significantly in both in-plane and out-of-plane directions.