Please confer Corollary 2.2 of the following with $d$ equal to $3$ and with $n$ equal to $2$.

Jason Starr
A pencil of Enriques surfaces of index 1 with no section
https://arxiv.org/pdf/math/0602639.pdf

This proves that for every integer $d\geq 2$, for a very general hypersurface $X$ in $\mathbb{P}^2\times \mathbb{P}^1$ in the complete linear system of $\mathcal{O}_{\mathbb{P}^2\times \mathbb{P}^1}(3,d)$, there is no rational section of the projection, $$\text{pr}_2|_X:X\to \mathbb{P}^1.$$ Since the locus of multiple curves in the complete linear system $\mathcal{O}_{\mathbb{P}^2}(3)$ has codimension $3$, for a general $X$ in the complete linear system, there are no multiple fibers.

By adjunction, the dualizing sheaf of $X$ equals $\mathcal{O}_{\mathbb{P}^2\times \mathbb{P}^1}(0,d-2)|_X$. Thus, for $d\geq 3$, the dualizing sheaf is the pullback of an ample sheaf by $\text{pr}_2|_X$. Thus, the Kodaira dimension equals $1$.

Please confer Corollary 2.2 of the following with $d$ equal to $3$ and with $n$ equal to $2$.

Jason Starr
A pencil of Enriques surfaces of index 1 with no section
https://arxiv.org/pdf/math/0602639.pdf

This proves that for every integer $e\geq 2$, for a very general hypersurface $X$ in $\mathbb{P}^2\times \mathbb{P}^1$ in the complete linear system of $\mathcal{O}_{\mathbb{P}^2\times \mathbb{P}^1}(3,e)$, there is no rational section of the projection, $$\text{pr}_2|_X:X\to \mathbb{P}^1.$$ Since the locus of multiple curves in the complete linear system $\mathcal{O}_{\mathbb{P}^2}(3)$ has codimension $3$, for a general $X$ in the complete linear system, there are no multiple fibers.

By adjunction, the dualizing sheaf of $X$ equals $\mathcal{O}_{\mathbb{P}^2\times \mathbb{P}^1}(0,e-2)|_X$. Thus, for $e\geq 3$, the dualizing sheaf is the pullback of an ample sheaf by $\text{pr}_2|_X$. In that case, the Kodaira dimension equals $1$.