We consider the unweighted bipartite maximum matching problem in the one-pass turnstile streaming model where the input stream consists of edge insertions and deletions. In the insertion-only model, a one-pass $2$-approximation streaming algorithm can be easily obtained with space $O(n \log n)$, where $n$ denotes the number of vertices of the input graph. We show that no such result is possible if edge deletions are allowed, even if space $O(n^{3/2-δ})$ is granted, for every $δ> 0$. Specifically, for every $0 \le ε\le 1$, we show that in the one-pass turnstile streaming model, in order to compute a $O(n^ε)$-approximation, space $Ω(n^{3/2 - 4ε})$ is required for constant error randomized algorithms, and, up to logarithmic factors, space $O( n^{2-2ε} )$ is sufficient. Our lower bound result is proved in the simultaneous message model of communication and may be of independent interest.