Let \$C\$ be an irreducible, smooth, projective curve of genus \$g$\backslash$geqslant 3\$ over the complex field \$$\backslash$mathbb\{C\}\$. The curve \$C\$ is called \{$\backslash$em bielliptic\} if it admits a degree-two morphism \$$\backslash$pi$\backslash$colon C$\backslash$longrightarrow E\$ onto an elliptic curve \$E\$; such a morphism is called a \{$\backslash$em bielliptic structure\} on \$C\$. If \$C\$ is bielliptic and \$g$\backslash$geqslant 6\$, then the bielliptic structure on \$C\$ is unique, but if \$g=3,4,5\$, then this holds true only generically and there are curves carrying \$n>1\$ bielliptic structures. The sharp bounds \$n$\backslash$leqslant 21,10,5\$ exist for \$g=3,4,5\$ respectively. Let \$$\backslash$mathfrak\{M\}\_g\$ be the coarse moduli space of irreducible, smooth, projective curves of genus \$g=3,4,5\$. Denote by \$$\backslash$mathfrak\{B\}\_g{\^{}}n\$ the locus of points in \$$\backslash$mathfrak\{M\}\_g\$ representing curves carrying at least \$n\$ bielliptic structures. It is then natural to ask the following questions. Clearly \$$\backslash$mathfrak\{B\}\_g{\^{}}n$\backslash$subseteq $\backslash$mathfrak\{B\}\_g{\^{}}\{n-1\}\$; does \$$\backslash$mathfrak\{B\}\_g{\^{}}n$\backslash$neq$\backslash$mathfrak\{B\}\_g{\^{}}\{n-1\}\$ hold? What are the irreducible components of \$$\backslash$mathfrak\{B\}\_g{\^{}}n\$? Are the irreducible components of \$$\backslash$mathfrak\{B\}\_g{\^{}}n\$ rational? How do the irreducible components of \$$\backslash$mathfrak\{B\}\_g{\^{}}n\$ intersect each other? Let \$C$\backslash$in$\backslash$mathfrak\{B\}\_g{\^{}}2\$; how many non-isomorphic elliptic quotients can it have? Complete answers are given to the above questions in the case \$g=4\$.