In the literature, there are various methods to obtain the fractal sets such as escape time algorithm, L-systems and iterated function system (IFS), etc. In this study, we aim to approximate to the classical fractals by using non-affine contraction mappings. In order to get these non-affine mappings, we utilize from the sequences of suitable Lipschitz continuous functions. Then, we obtain some approximations to the fractals which can be constructed as the attractor of an IFS. Finally, we give some illustrations for some specific cases.