Superconductivity (SC) in so-called “unconventional superconductors” is nearly always found in the vicinity of another ordered state, such as antiferromagnetism, charge density wave (CDW), or "stripe" order. This suggests a fundamental connection between SC and fluctuations in some other order parameter. In this talk I will describe our use of high-pressure x-ray scattering to directly study the CDW order in the layered dichalcogenide 1T-TiSe₂, which was previously shown to exhibit SC when the CDW is suppressed by pressure or intercalation of Cu atoms. We succeeded in suppressing the CDW fully to zero temperature, establishing for the first time the existence of a quantum critical point (QCP) at P_c = 5.1 ± 0.2 GPa, which is more than 1 GPa beyond the end of the SC dome. Unexpectedly, at P = 3 GPa we observed a reentrant, weakly first order, incommensurate phase, indicating the presence of a Lifshitz tricritical point somewhere above the superconducting dome. Our study suggests that SC in TiSe₂ may not be connected to the amplitude QCP itself, but to the formation of CDW domain walls.