The contribution of ocean fronts to the properties and temporal evolution of Sea Surface Temperature (SST) structure functions have been investigated using a numerical model of the California Current system. First, the intensity of fronts have been quantified by using singularity exponents. Then, leaning on the multifractal theory of turbulence, we show that the departure of the scaling of the structure functions from a straight line, known as anomalous scaling, depends on the intensity of the strongest fronts. These fronts, at their turn, are closely related to the seasonal change of intensity of the coastal upwelling characteristics of this area. Our study points to the need to correctly reproduce the intensity of the strongest fronts and, consequently, properly model processes such as coastal upwelling in order to reproduce SST statistics in ocean models.