  For a given graph $F$, the $F$-saturation number of a graph $G$, denoted by $
{sat}(G, F)$, is the minimum number of edges in an edge-maximal $F$-free
subgraph of $G$. In 2017, Kor\'andi and Sudakov determined $ {sat}({G}(n, p),
K_r)$ asymptotically, where ${G}(n, p) $ denotes the Erd\H{o}s-R\'enyi random
graph and $ K_r$ is the complete graph on $r$ vertices. In this paper, among
other results, we present an asymptotic upper bound on ${sat}({G}(n, p), F)$
for any bipartite graph $F$ and also an asymptotic lower bound on ${sat}({G}(n,
p), F)$ for any complete bipartite graph $F$.
