  We use the circle method to count the number of integer solutions to systems
of bihomogeneous equations of bidegree $(1,1)$ and $(2,1)$ of bounded height in
lopsided boxes. Previously, adjusting Birch's techniques to the bihomogeneous
setting, Schindler showed an asymptotic formula provided the number of
variables grows at least quadratically with the number of equations considered.
Based on recent methods by Rydin Myerson we weaken this assumption and show
that the number of variables only needs to satisfy a linear bound in terms of
the number of equations.
