The theory of fewnomials seeks quantitative bounds on polynomial systems in terms of the number of nonzero monomials occurring in the system. These bounds can be of different nature. The first focus of the fewnomial theory was to find bounds for the number of real solutions of multivariate sparse systems. Nevertheless, one can also ask for bounds on the maximal multiplicity of a complex solution to a sparse polynomial system. In this work, we study bivariate systems defined by two curves. We consider a mixed model in which one curve has a bounded number of monomials, while the other has a bounded degree. We show that the intersection multiplicity of any isolated solution of such system is polynomially bounded by these two parameters, provided that the solution has nonzero coordinates. This is similar to the real case, since an analogous bound is known for the number of real solutions of these types of systems. We also discuss the connections between sparse polynomials and algebraic complexity theory. This is joint work with Pascal Koiran.