The quasi-one-dimensional topological insulator ${\mathrm{TaSe}}_{3}$ offers an intriguing platform to study the intertwined exotic quantum order, such as superconductivity and charge-density-wave (CDW) order, and nontrivial topological states. Nevertheless, it remains unclear how robust the superconductivity in this quasi-one-dimensional material is and how close the (hidden) CDW exists in its phase diagram. In this work, we unveil a well-defined CDW transition in ${\mathrm{TaSe}}_{3}$ crystals by leveraging a hydrostatic pressure. Remarkably, it is found that the CDW transition temperature gradually strengthens as the pressure increases, and crosses the room temperature above $\ensuremath{\sim}2\phantom{\rule{0.16em}{0ex}}\mathrm{GPa}$, which makes it promising in future applications in electronic devices. Furthermore, based on the pressure-dependent resistivity and Raman spectroscopy measurements by using a diamond anvil cell, we have extended the high-pressure study up to 50 GPa and verified a proposed phase transition around 6--10 GPa. However, no superconductivity is observed down to 2 K in the whole pressure range studied, suggesting the superconductivity is fragile in this quasi-one-dimensional transition-metal trichalcogenide.