Enhancement of superconductivity on the verge of a structural instability in isovalently doped $\beta$-ThRh$_{1-x}$Ir$_{x}$Ge

$\beta$-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.