Structural quality of LuFeO3 epitaxial layers grown by pulsed-laser deposition on sapphire substrates with and without platinum Pt interlayers has been investigated by in situ high-resolution X-ray diffraction (reciprocal-space mapping). The parameters of the structure such as size and misorientation of mosaic blocks have been determined as functions of the thickness of LuFeO3 during growth and for different thicknesses of platinum interlayers up to 40 nm. By means of fitting of the time-resolved X-ray reflectivity curves and by in situ X-ray diffraction measurement, we demonstrate that the LuFeO3 growth rate as well as the out-of-plane lattice parameter are almost independent from Pt interlayer thickness, while the in-plane LuFeO3 lattice parameter decreases. We reveal that, despite the different morphologies of the Pt interlayers with different thickness, LuFeO3 was growing as a continuous mosaic layer and the misorientation of the mosaic blocks decreases with increasing Pt thickness. The X-ray diffraction results combined with ex situ scanning electron microscopy and high-resolution transmission electron microscopy demonstrate that the Pt interlayer significantly improves the structure of LuFeO3 by reducing the misfit of the LuFeO3 lattice with respect to the material underneath.