  $\beta$-ThRhGe, the high-temperature polymorph of ThRhGe, is isostructural to
the well-known ferromagnetic superconductor URhGe. However, contrary to URhGe,
$\beta$-ThRhGe is nonmagnetic and undergoes an incomplete structural phase
transition at 244 K, followed by a superconducting transition below 3.36 K.
Here we show that the isovalent substitution of Ir for Rh leads to a strong
enhancement of superconductivity by suppressing the structural transition. At
$x$ = 0.5, where the structural transition disappears, $T_{\rm c}$ reaches a
maximum of 6.88 K. The enhancement of superconductivity is linked to the
proximity to a structural quantum critical point at this Ir concentration, as
suggested by the analysis of thermodynamic as well as resistivity data. First
principles calculations indicate that the Ir doping has little effect on the
electronic band dispersion near the Fermi level. $\beta$-ThRh$_{1-x}$Ir$_{x}$Ge
thus provides an excellent platform to study the interplay between
superconductivity and structural quantum criticality in actinide-containing
compounds.
