  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.
