On a degree $n$ Hirzebruch surface $F_n$, suppose we have a very ample linear system. It is known that its generic smooth irreducible members give a Lefschetz pencil on $F_n$. Let us take a member, $G$, in this pencil. And suppose we know that $G$ intersects the $-n$ section, $S$, of $F_n$ 4 times. Generically they intersect at 4 points. 

Is it possible to move $G$ in a way that $G$ and $S$ intersect at 1 point with multiplicity 4? I don't want to change the genus of $G$ while moving. Or is it possible to find $G'$ in the pencil which intersects $S$ in the desired way?

I am not an algebro-geometer so, sorry if this question is trivial. I'd appreciate any suggestions. Thanks.