  We study the space-time nonlinear fractional stochastic heat equation driven
by a space-time white noise,
  \begin{align*}
  \partial_t^\beta
u(t,x)=-(-\Delta)^{\alpha/2}u(t,x)+I_t^{1-\beta}\Big[\sigma(u(t,x))\dot{W}(t,x)\Big],\
\ t>0, \ x\in \mathbb{R} ,
  \end{align*}
  where $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ is a globally Lipschitz
function and the initial condition is a measure on $\mathbb{R}.$ Under some
growth conditions on $\sigma,$ 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 $x\in\mathbb{R}^d.$
