In full generality, this is false.
I'll construct an example in the blow up $X$ of $\mathbb{CP}^2$ (which is, in fact, an $S^2$-bundle over $S^2$).
Kronheimer and Mrowka proved that every symplectic surface is genus-minimising in its homology class, and Severi proved that there are two irreducible plane complex curves $C_1, C_2\subset\mathbb{CP}^2$ of degree 4 with one and three nodes respectively. Notice that $g(C_1) = 2$, while $C_2$ is rational (i.e. it's an immersed sphere).
If you blow them up at one of their nodes, you get two symplectic surfaces in (a copy of) $X$, in the same homology class (namely, $4h-2e$, where $h$ is the line class and $e$ is the class of the exceptional divisor): the proper transform of $C_1$ is embedded and symplectic, so it's also genus-minimising in its homology class; on the other hand, the proper transform of $C_2$ is an immersed sphere (with two double points), so it has lower genus.
I would assume that similar counterexamples can be found for most (all?) surface bundles over surfaces, but I wouldn't swear.
In general, the minimal genus problem is a quite hard one, and the only answers I can think of come from gauge theory (Seiberg-Witten or Heegaard Floer).
