We study the space-time nonlinear fractional stochastic heat equation driven by a space-time white noise, \begin{align*} \partial_t^βu(t,x)=-(-Δ)^{α/2}u(t,x)+I_t^{1-β}\Big[σ(u(t,x))\dot{W}(t,x)\Big],\ \ t>0, \ x\in \mathbb{R} , \end{align*} where $σ:\mathbb{R}\rightarrow\mathbb{R}$ is a globally Lipschitz function and the initial condition is a measure on $\mathbb{R}.$ Under some growth conditions on $σ,$ we derive two important properties about the moments of the solution: (i) For $p\geq 2,$ the $p^{\text{th}}$ absolute moment of the solution to the equation above grows exponentially with time. (ii) Moreover, the distances to the origin of the farthest high peaks of these moments grow exactly exponentially with time. Our results provide an extension of the work of Chen and Dalang (Stoch PDE: Anal Comp (2015) 3:360-397) to a time-fractional setting. We also show that condition (i) holds when we study the same equation for $x\in\mathbb{R}^d.$