On some fibrations of $\overline{M}_{0,n}$

The paper is a second step in the study of $\overline{M}_{0,n}$ started in arXiv:1006.0987 [math.AG]. We study fiber type morphisms of this moduli space via Kapranov's beautiful description. Our final goal is to understand if any dominant morphism $f: \overline{M}_{0,n} \to X$ with positive dimensional fibers factors through forgetful morphisms. We prove that this is true if either $n \leq 7$ or $\rm {dim} X \leq 2$ or the rational map induced on $P^{n-3}$ has linear general fibers. Along the way we give examples of forgetful morphisms whose fibers are connected curves of arbitrarily high positive genus, for $n>>0$.