We prove that for every ${γ> 0}$ there exists $n_0 \in \mathbb{N}$ such that for every ${n \geq n_0}$ any family of up to $\lfloor{n^{\frac12+γ}}\rfloor$ trees having at most $(1-γ)n$ vertices in each bipartition class can be packed into $K_{n,n}$. As a tool for our proof, we show an approximate bipartite version of the Komlós-Sárközy-Szemerédi Theorem, which we believe to be of independent interest.