There has been a concerted effort in recent years to identify the astrophysical sites of the r-process that can operate early in the galaxy. The discovery of many r-process-enhanced (RPE) stars (especially by the R-process Alliance collaboration) has significantly accelerated this effort. However, only limited data exist on the detailed elemental abundances covering the primary neutron-capture peaks. Subtle differences in the structure of the r-process pattern, such as the relative abundances of elements in the third peak, in particular, are expected to constrain the r-process sites further. Here, we present a detailed elemental-abundance analysis of four bright RPE stars selected from the HESP-GOMPA survey. Observations were carried out with the 10-m class telescope Gran Telescopio Canarias (GTC), Spain. The high spectral signal-to-noise ratios obtained allow us to derive abundances for 20 neutron-capture elements, including the third r-process peak element osmium (Os). We detect thorium (Th) in two stars, which we use to estimate their ages. We discuss the metallicity evolution of Mg, Sr, Ba, Eu, Os, and Th in r-II and r-I stars, based on a compilation of RPE stars from the literature. The strontium (Sr) abundance trend with respect to europium (Eu) suggests the need for an additional production site for Sr (similar to several earlier studies); this requirement could be milder for yttrium (Y) and zirconium (Zr). We also show that there could be some time delay between r-II and r-I star formation, based on the Mg/Th abundance ratios.