We study the contribution of confined phonons to the polaron energy in quantum-well wires. We use a dispersionless, macroscopic continuum model to describe the phonon confinement in quantum wires of square cross section. The polaron energy is calculated variationally incorporating the dynamic screening effects. We find that the confined phonon contribution to the polaron energy is comparable to that of bulk phonons in the density range N = 104-107 cm-1. Screening effects within the random-phase approximation significantly reduce the electron-confined phonon interaction, whereas the correlation effects tend to oppose this trend.