For given graphs $G$ and $F$, the Turán 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 Turán 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 Turán 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 Turán number of $C_4$ and $P_{\ell}$, suggesting that in the latter problem the asymptotic behavior depends heavily on the parity of $\ell$.