I may have missed something but it should follow from Bonnesen's inequality, which states that every domain $\Omega\subset\mathbb{R}^2$ satisfies : $$\mathcal{L}(\partial\Omega)^2-4\pi\mathcal{A}(\Omega)\geq \pi^2(r_\text{ex}(\Omega)-r_\text{in}(\Omega))^2 $$ where $r_\text{in}(\Omega)$ (resp. $r_\text{ex}(\Omega)$) is the smallest (rest. biggest) possible radius of disk contained in $\bar\Omega$ (resp. which contains $\Omega$).

If one denotes by $\Omega_n$ the domain bounded by your $\gamma_n$. Then your hypothesis imply that each of the $\Omega_n$ are sandwiched between two disks of closer and closer radius, which should be enough to conclude.

I may have missed something but it should follow from Bonnesen's inequality, which states that every domain $\Omega\subset\mathbb{R}^2$ satisfies : $$\mathcal{L}(\partial\Omega)^2-4\pi\mathcal{A}(\Omega)\geq \pi^2(r_\text{ex}(\Omega)-r_\text{in}(\Omega))^2 $$ where $r_\text{in}(\Omega)$ (resp. $r_\text{ex}(\Omega)$) is the biggest( resp. smallest) possible radius of disk contained in $\Omega$ (resp. which contains $\bar\Omega$).

If one denotes by $\Omega_n$ the domain bounded by your $\gamma_n$. Then your hypothesis imply that each of the $\Omega_n$ are sandwiched between two disks of closer and closer radius, which should be enough to conclude.