Consider a rectifiable Jordan curve $\Gamma$ and let $\gamma: S^1 \rightarrow \Gamma$ be a homeomorphism. Moreover, let $\Omega\subset\mathbb{R}^2 \setminus\Gamma$ be the interior of $\Gamma$. For each $\varepsilon>0$, is there a Jordan polygon $P \subset \Omega \cup \Gamma$ (which need not be inscribed in $\Gamma$, although preferably it should be) and a homeomorphism $\rho: S^1 \rightarrow P$ such that
$$\max_{t \in S^1} |\gamma(t) - \rho(t)|< \varepsilon \quad \text{and} \quad |L(\Gamma)-L(P)|< \varepsilon$$
where $L$ stands for length? If we abolish the condition $P \subset \Omega \cup \Gamma$, then the answer is yes. However, with this condition I am not sure if this holds true.