Positive discrepancy of graphs far from Turán graphs

We prove that, for every $\text{eps}>0$, any $n$-vertex graph that needs at least $\text{eps}{n}^2$ edge changes to become a Turán graph has positive discrepancy at least $c_{\text{eps}}{n}^{5\mathrm{/}4}$. Consequently, every such regular graph has second eigenvalue at least $c_{\text{eps}}^{\mathrm{\prime }}{n}^{1\mathrm{/}4}$. These results prove two conjectures of Räty, Sudakov and Tomon.