  ThFeAsN$_{1-x}$O$_x$ ($0\leq x\leq0.6$) system with heavy electron doping has
been studied by the measurements of X-ray diffraction, electrical resistivity,
magnetic susceptibility and specific heat. The non-doped compound exhibits
superconductivity at $T_\mathrm{c}^\mathrm{onset}=30$ K, which is possibly due
to an internal uniaxial chemical pressure that is manifested by the extremely
small value of As height with respect to the Fe plane. With the oxygen
substitution, the $T_\mathrm{c}$ value decreases rapidly to below 2 K for
$0.1\leq x\leq0.2$, and surprisingly, superconductivity re-appears in the range
of $0.25\leq x\leq0.5$ with a maximum $T_\mathrm{c}^\mathrm{onset}$ of 17.5 K
at $x=0.3$. For the normal-state resistivity, while the samples in intermediate
non-superconducting interval exhibit Fermi liquid behavior, those in other
regions show a non-Fermi-liquid behavior. The specific heat jump for the
superconducting sample of $x=0.4$ is $\Delta C/(\gamma T_\mathrm{c})=0.89$,
which is discussed in terms of anisotropic superconducting gap. The peculiar
phase diagram in ThFeAsN$_{1-x}$O$_x$ presents additional ingredients for
understanding the superconducting mechanism in iron-based superconductors.
