  We study existence of solutions for the fractional problem \begin{equation*}
(P_m) \quad \left \{
  \begin{aligned}
  (-\Delta)^{s} u + \mu u &=g(u) & \; \text{in $\mathbb{R}^N$}, \cr
  \int_{\mathbb{R}^N} u^2 dx &= m, & \cr
  u \in H^s_r&(\mathbb{R}^N), &
  \end{aligned}
  \right. \label{problemx} \end{equation*} where $N\geq 2$, $s\in (0,1)$,
$m>0$, $\mu$ is an unknown Lagrange multiplier and $g \in C(\mathbb{R},
\mathbb{R})$ satisfies Berestycki-Lions type conditions. Using a Lagrange
formulation of the problem $(P_m)$, we prove the existence of a weak solution
with prescribed mass when $g$ has $L^2$ subcritical growth. The approach relies
on the construction of a minimax structure, by means of a Pohozaev's mountain
in a product space and some deformation arguments under a new version of the
Palais-Smale condition introduced in [16,19]. A multiplicity result of
infinitely many normalized solutions is also obtained if $g$ is odd.
