The bipartite Turán number of a graph H, denoted by ex(m,n;H), is the maximum number of edges in any bipartite graph G=(X,Y;E) with |X|=m and |Y|=n which does not contain H as a subgraph. In this paper, we determine ex(m,n;Fℓ) for arbitrary ℓ and appropriately large n in comparison to m and ℓ, where Fℓ is a linear forest which consists of ℓ vertex disjoint paths. Moreover, the extremal graphs have been characterized. Furthermore, these results are used to obtain the maximum spectral radius of bipartite graphs which do not contain Fℓ as a subgraph and characterize all extremal graphs which attain the maximum spectral radius.