  Using a data sample corresponding to an integrated luminosity of 11.3 $\rm
fb^{-1}$ collected at center-of-mass energies from $4.23$ to $4.70$ GeV with
the BESIII detector, we measure the product of the $e^+e^-\to
\pi^+\pi^-\psi_2(3823)$ cross section and the branching fraction
$\mathcal{B}[\psi_2(3823)\to \gamma\chi_{c1}]$.
  For the first time, resonance structure is observed in the cross section line
shape of $e^+e^-\to \pi^+\pi^-\psi_2(3823)$ with significances exceeding
$5\sigma$.
  A fit to data with two coherent Breit-Wigner resonances modeling the
$\sqrt{s}$-dependent cross section yields $M(R_1)=4406.9\pm 17.2\pm 4.5$
MeV/$c^2$, $\Gamma(R_1)=128.1\pm 37.2\pm 2.3$ MeV, and $M(R_2)=4647.9\pm 8.6\pm
0.8$ MeV/$c^2$, $\Gamma(R_2)=33.1\pm 18.6\pm 4.1$ MeV.
  Though weakly disfavored by the data, a single resonance with
$M(R)=4417.5\pm26.2\pm3.5$ MeV/$c^2$, $\Gamma(R)=245\pm48\pm13$ MeV is also
possible to interpret data.
  This observation deepens our understanding of the nature of the vector
charmoniumlike states.
  The mass of the $\psi_2(3823)$ state is measured as $(3823.12\pm 0.43\pm
0.13)$ MeV/$c^2$, which is the most precise measurement to date.
