  For given graphs $G$ and $F$, the Tur\'an number $ex(G,F)$ is defined to be
the maximum number of edges in an $F$-free subgraph of $G$. Foucaud,
Krivelevich and Perarnau and later independently Briggs and Cox introduced a
dual version of this problem wherein for a given number $k$, one maximizes the
number of edges in a host graph $G$ for which $ex(G,H) < k$.
  Addressing a problem of Briggs and Cox, we determine the asymptotic value of
the inverse Tur\'an number of the paths of length $4$ and $5$ and provide an
improved lower bound for all paths of even length. Moreover, we obtain bounds
on the inverse Tur\'an number of even cycles giving improved bounds on the
leading coefficient in the case of $C_4$. Finally, we give multiple conjectures
concerning the asymptotic value of the inverse Tur\'an number of $C_4$ and
$P_{\ell}$, suggesting that in the latter problem the asymptotic behavior
depends heavily on the parity of $\ell$.
