Studies of supported water clusters provide a means for understanding the initial stages of heterogeneous ice nucleation in diverse areas as atmospheric chemistry and astrophysics. Despite the importance of non-perfect ice structures in these fields, research focused on crystalline ice structures. Here, we report real-space observations of fractal ice islands grown between 89K and 119K. The island shape changes linearly from the most fractal dimension of 5/3 to the fractal dimension close to the one of an equal-sided hexagon. The mere linear increase is assigned to a shape dependent sticking coefficient of mobile ice clusters to the fractal islands. Our study reveals the complexity involved in formation of fractal ice structures.