  Flow harmonics ($v_n$) in the Fourier expansion of the azimuthal distribution
of particles are widely used to quantify the anisotropy in particle emission in
high-energy heavy-ion collisions. The symmetric cumulants, $SC(m,n)$, are used
to measure the correlations between different orders of flow harmonics. These
correlations are used to constrain the initial conditions and the transport
properties of the medium in theoretical models. In this Letter, we present the
first measurements of the four-particle symmetric cumulants in Au+Au collisions
at $\sqrt{s_{NN}}$ = 39 and 200 GeV from data collected by the STAR experiment
at RHIC. We observe that $v_{2}$ and $v_{3}$ are anti-correlated in all
centrality intervals with similar correlation strengths from 39 GeV Au+Au to
2.76 TeV Pb+Pb (measured by the ALICE experiment). The $v_{2}$-$v_{4}$
correlation seems to be stronger at 39 GeV than at higher collision energies.
The initial-stage anti-correlations between second and third order
eccentricities are sufficient to describe the measured correlations between
$v_{2}$ and $v_{3}$. The best description of $v_{2}$-$v_{4}$ correlations at
$\sqrt{s_{NN}}$ = 200 GeV is obtained with inclusion of the system's nonlinear
response to initial eccentricities accompanied by the viscous effect with
$\eta/s$ $>$ 0.08. Theoretical calculations using different initial conditions,
equations of state and viscous coefficients need to be further explored to
extract $\eta/s$ of the medium created at RHIC.
