Large-amplitude decaying kink oscillations of coronal loops are strongly influenced by nonlinear processes such as Kelvin─Helmholtz instability (KHI) and turbulence, although comprehensive theoretical understanding and observational confirmation remain limited. Building on the recently developed theory on nonlinear damping by KHI-induced turbulence in impulsively driven transverse loop oscillations, we investigate its observational signatures using three-dimensional magnetohydrodynamic simulations and forward-modeled extreme-ultraviolet images. The simulated oscillations exhibit time-varying frequency shifts and damping rates, which are broadly consistent with nonlinear turbulence-damping theory. In addition, they exhibit excitation of higher-order modes, slightly increased periods relative to the linear kink period, and reduced displacement amplitudes. These features are generally preserved in synthetic observations, though resolving higher-order modes requires higher spatial resolution than is currently available. For loops embedded in a hotter background, hotter channels (e.g., 193 Å) are more sensitive to boundary dynamics; consequently, their oscillations decay faster with smaller displacements and larger phase shifts than those in cooler channels (e.g., 171 Å). Comparisons of simulated and synthetic oscillations show close agreement at the early stage. At later times, synthetic oscillations exhibit smaller displacements and larger phase shifts, due to turbulence-induced asymmetry in the loop cross section. Bayesian fitting shows that the initial oscillation amplitude and kink period are robustly constrained, whereas parameters controlling the damping profile are degenerate, indicating that additional observables would aid reliable seismological inference. These results provide a quantitative basis for identifying nonlinear damping and detecting KHI-driven turbulence in transverse loop oscillations.