Packing large balanced trees into bipartite graphs

We prove that for every $\gamma >0$ there exists $n_0\in \mathbb{N}$ such that for every $n\ge n_0$ any family of up to $\lfloor n^{\frac{1}{2}+\gamma }\rfloor$ trees having at most $(1-\gamma )n$ 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.