The impacts of four distinctive types of violent breaking waves on a flexible wall at a large scale are investigated using a fully-coupled computational fluid dynamics (CFD) and computational solid mechanics (CSM) model in the finite-volume framework, in which the CFD model simulates the incompressible two-phase (water-air) flow, and the CSM model considers a Neo-Hookean solid. The model is well validated against two experiments of the breaking wave impact on a rigid wall and the dam-break impact on an elastic wall. We then apply this model to study the interaction between the breaking waves and an elastic wall. Four types of breaking wave impact in terms of the slightly-breaking, low-aeration, high-aeration, and broken-wave impacts are considered and Cauchy number of the flexible wall ranges from 0.72 to 1.30. Compared with the rigid wall, it is found that the profile of peak pressure on the flexible wall shifts slightly upward. The impact force, impact duration, and impact impulse are affected (not necessarily reduced) by the structural deformation in progressive waves. The von Mises stress in the wall shows that the structural integrity of rigid and flexible walls is susceptible to the impact and maximum quasi-hydrostatic forces, respectively. Under the maximum quasi-hydrostatic force, the peak displacement of the flexible wall appears with a magnitude from 0.21 to 0.48 wave excursion, which exacerbates the wave overtopping and the von Mises stress in the wall for the tested impacts. Afterward, the flexible wall vibrates at a frequency very close to its natural frequency in vacuo, which is independent of the incident wave frequency (being different from the non-breaking periodic wave-induced motions). Finally, the effect of air compressibility for simulating the violent breaking wave impact with considerable air entrapment at a large scale is discussed. The findings in this study can support the design of existing and novel offshore and coastal structures with flexibility.