We investigate the adhesion dynamics of flexible elastic structures contacting a rigid surface coated with a viscous fluid and retracted at a controlled speed, mimicking natural adhesive mechanisms. During detachment by cohesive failure, the confined viscous fluid generates an adhesive force through a pressure drop induced by flow within the thin film. Dedicated experiments reveal that structural flexibility gives rise to a strongly nonlinear mechanical response, deeply affecting both the magnitude of the adhesion force and its dependence on the retraction velocity. Unlike rigid systems, flexible structures enable adhesion to be either enhanced or reduced simply by varying the withdrawal speed or the elasticity of the structures, providing a direct path for tuning adhesive performance. To capture this behavior, we develop a theoretical framework that couples elasticity and viscosity, offering new insights into the role of structural flexibility in controlling viscous adhesion.