We describe the localized folding of thick layers embedded in a viscoelastic framework. Higher-order partial differential equations such as the Swift-Hohenberg equation are standard for modelling the folding process. Using a high-order shear theory, we modify the Swift-Hohenberg equation to describe the buckling of thick layers and consider the folded layer's viscoelastic behavior. The use of thick layers enables us to consider shear strains parallel to the layers during folding. Our model naturally captures the softening-stiffening behavior by including a non-linear viscoelastic description using a Winkler-type foundation. Next, we study the linear stability behavior of the system and derive the dispersion relations. Finally, we simulate this new model using a robust custom-built isogeometric analysis solver, which allows us to describe thick folded layers with localized folding. The numerical results show that the folding of an elastic layer produces periodic patterns while a viscoelastic layer deflects locally. When the horizontal forces are unequal, the periodic folds initiate in the direction of the smaller force and the localized deformations occur parallel to the larger force. Later, the nonlocal deformation occurs in the direction of a smaller force. Domes and basins or more linear ridges and valleys are formed according to the relative magnitudes of the applied forces. Domes and basins are the results of equal horizontal applied forces, and non-equal forces result in ridges and valleys.