Layered textit{Ln}OBiS$_2$ compounds with textit{Ln} = La, Ce, Pr, Nd, and Yb can be rendered conducting and superconducting via two routes, substitution of F for O or the tetravalent ions Ti, Zr, Hf, and Th for trivalent textit{Ln} ions. Electrical resistivity measurements on non-fluorinated La$_{0.80}$Ti$_{0.20}$OBiS$_2$ and La$_{0.85}$Th$_{0.15}$OBiS$_2$ superconductors were performed between $sim$1.5 K and 300 K and under pressure up to 2.4 GPa. For both compounds, the superconducting transition temperature $T_c$, which is $sim$2.9 K at ambient pressure, gradually increases with pressure to 3.2-3.7 K at $sim$1 GPa, above which it is suppressed and the superconducting transitions become very broad. Measurements of the normal state electrical resistivity of the two compounds reveal discontinuous changes of the resistivity as a function of pressure at $sim$0.6 GPa. Surprisingly, above 1.3 GPa, semiconducting-like behavior reappears in La$_{0.80}$Ti$_{0.20}$OBiS$_2$. This study reveals a new high-pressure phase of La$_{1-x}$$M$$_x$OBiS$_2$ containing the tetravalent ions $M$ = Ti, Th which does not favor superconductivity. In contrast, application of pressure to fluorinated LaO$_0.5$F$_0.5$BiS$_2$ produces an abrupt tetragonal-monoclinic transition to a metallic phase with an enhanced $T_c$. These results demonstrate that the response of the normal and superconducting properties of LaOBiS$_2$-based compounds depends strongly on the atomic site where the electron donor ions are substituted.
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