  We consider a branching random walk on $\mathbb{Z}$ started by $n$ particles
at the origin, where each particle disperses according to a mean-zero random
walk with bounded support and reproduces with mean number of offspring
$1+\theta/n$. For $t\geq 0$, we study $M_{nt}$, the rightmost position reached
by the branching random walk up to generation $[nt]$. Under certain moment
assumptions on the branching law, we prove that $M_{nt}/\sqrt{n}$ converges
weakly to the rightmost support point of the local time of the limiting
super-Brownian motion. The convergence result establishes a sharp exponential
decay of the tail distribution of $M_{nt}$. We also confirm that when
$\theta>0$, the support of the branching random walk grows in a linear speed
that is identical to that of the limiting super-Brownian motion which was
studied by Pinsky in [28]. The rightmost position over all generations,
$M:=\sup_t M_{nt}$, is also shown to converge weakly to that of the limiting
super-Brownian motion, whose tail is found to decay like a Gumbel distribution
when $\theta<0$.
