For links $L\subset \mathrm{\Sigma }\times [0,1]$, where $\mathrm{\Sigma }$ is a closed orientable surface, we define a $U_q(\mathfrak{gl}(1,1))$ Reshetikhin-Turaev invariant with coefficients in $\mathbb{Z}[H_1(\mathrm{\Sigma })]$. This is well-defined up to multiples of the quantum supergroup variable $q$. This invariant turns out to be equivalent to an infinite cyclic version of the Carter-Silver-Williams (CSW) polynomial. The importance of the CSW polynomial is that half its symplectic rank gives strong lower bounds on the virtual genus. Recall that given a virtual link type $L$, the virtual genus of $L$ is the smallest genus of all closed orientable surfaces $\mathrm{\Sigma }$ on which $L$ can be represented by a diagram $D$ on $\mathrm{\Sigma }$. The main objective of this paper is to extend the CSW bound on the virtual genus to all Lie superalgebras $U_q(\mathfrak{gl}(m,n))$ with $n>0$. For links in thickened once-punctured surfaces $\mathrm{\Sigma }$, we define a $U_q(\mathfrak{gl}(m,n))$ Reshetikhin-Turaev invariant with coefficients in $\mathbb{Z}[H_1(\mathrm{\Sigma })]$. We show that half its symplectic rank is also a lower bound on the virtual genus. Changing the value of the pair $(m,n)$ can give lower bounds better than those available from other known methods. We compare the $U_q(\mathfrak{gl}(m,n))$ lower bounds to those coming from the CSW polynomial, the surface bracket, the arrow polynomial, hyperbolicity, and the Gordon-Litherland determinant test. As an application, we show that the Seifert genus of homologically trivial knots in thickened surfaces is not additive under the connected sum operation of virtual knots. This is joint work with Killian Davis and Anup Poudel.