Summary: Let \(\chi_q\) denotes the real character modulo \(q\) whenever it exists. For \(X\) and \(Y\) large enough, we prove \[ \sum_{|q| \leq X}\biggl| \sum_{n \leq Y} \chi_q(n)\biggr|^4 \ll XY^2 X^\varepsilon \] for any \(\varepsilon>0\) and where the implied constant depends on \(\varepsilon\). This estimate is essentially best possible and had been conjectured by \textit{M. Jutila} in 1973 [Tr. Mat. Inst. Steklova 132, 247--250 (1973; Zbl 0281.10014)]. This proof is based on \textit{D. R. Heath-Brown}'s estimate for the mean square of the real character sums [see Acta Arith. 72, 235--275 (1995; Zbl 0828.11040), Corollary 2] and the reflection principle.