We employ the flux-flow instability (FFI) technique – a widely used method for determining the quasiparticle relaxation time (${\tau }_E$) – to investigate single-crystal superconducting titanium nitride (TiN) films characterized by weak vortex pinning and slow electron-phonon relaxation. By systematically varying the TiN strip width, we show that reliable extraction of the vortex velocity and ${\tau }_E$ is achievable only when the strip width exceeds the electron-phonon relaxation length. A comparative analysis of ${\tau }_E$ values obtained using various FFI models and other experimental methods reveals that quasiparticle relaxation within the vortex core is primarily governed by an increase in quasiparticle temperature. Our analysis provides a reliable way to determine ${\tau }_E$ from FFI measurements and may also be relevant to other superconducting systems in which fast electron-electron scattering is expected to lead to a thermal-like quasiparticle distribution function.