- In the current paper we show that the dimension of a family $ V $ of irreducible reduced curves in a given ample linear system on a toric surface $ S $ over an algebraically closed field is bounded from above by $-K_S. C+ p_g (C)-1$, where $ C $ denotes a general curve in the family. This result generalizes a famous theorem of Zariski to the case of positive characteristic. We also explore new phenomena that occur in positive characteristic: We show that the equality $\dim (V)=-K_S. C+ p_g (C)-1$ does not imply the nodality of $ C $ even if $ C $ belongs to the smooth locus of $ S $, and construct reducible Severi varieties on weighted projective planes in positive characteristic, parameterizing irreducible reduced curves of given geometric genus in a given ample linear system.