Suppose that $\pi : X \to S$ is an elliptic fibration, and that the fiber over a point $s \in S$ is reducible, with several components. Call one of them $C$. How can I compute the normal bundle of $C$ in $X$? I'm not assuming the base is a curve.

I'd be happy to hear how to do this in an example, so for concreteness let's say that $S$ is the versal deformation space of a fiber of Kodaira type $I_2$: the central fiber consists of two smooth rational curves meeting at two points, $S$ is two-dimensional, and there are two curves in the base through $s$ over which the fibers are nodal cubics, corresponding to smoothing the two nodes of the singular fiber. What are the normal bundles of the two curves in the singular fiber?

I can choose a smooth curve $\gamma \subset S$ going through $s$, with the preimage a smooth surface $X_\gamma \subset X$. It's easy to compute the normal bundle of $C$ in $X_\gamma$ (they are $(-2)$-curves in the example), and there is $0 \to N_{C/X_\gamma} \to N_{C/X}$, but it's not clear to me that I should expect $N_{C/X_\gamma}$ to be a direct summand, since $C$ isn't obtained as the complete intersection of such surfaces.