We have investigated how a strong magnetic field (${\bf B}$)
could decipher the anisotropic interaction in heavy quark ($Q$) and antiquark
($\bar Q$) bound states through the perturbative thermal QCD
in real-time formalism. So we thermalize the Schwinger propagator for
quarks in the lowest Landau level and the Feynman propagator
for gluons to calculate the gluon self-energy up to one loop
for massless flavours. For the quark-loop contribution to
the self-energy, the medium does not have any temperature correction and the
vacuum term gives rise an anisotropic term whereas the gluon-loop
yields the temperature correction. This finding in quark-loop
contribution corroborates the equivalence of a massless QED in (1+1)-dimension
with the massless thermal QCD in strong magnetic field, which
(quark sector) is reduced to (1+1)-dimension (longitudinal).
This anisotropy in the self-energy is then
being translated into the permittivity of the medium, which now
behaves like a tensor. Thus the permittivity
of the medium in the momentum space makes the $Q \bar Q$ potential
in the coordinate space anisotropic in strong ${\bf B}$. As a matter of fact, the
potential for $Q \bar Q$-pairs
aligned transverse to ${\bf B}$ is more attractive than the parallel
alignment. However, the potential is always more attractive
due to the softening of the electric screening
mass whereas the (magnitude) imaginary-part of the potential
becomes smaller, compared to $B=0$. As a consequence, the binding
energies (B.E.) of the ground states of $c \bar c$ and
$b \bar b$ get increased and the widths ($\Gamma$)
get decreased compared to $B =0$. The above
medium modifications to the properties of $ Q \bar Q$ bound
states then facilitate to study their quasi-free
dissociation in the medium in a strong magnetic
field. The dissociation temperatures are estimated for $J/\psi$ and $\Upsilon$
states quantitatively
as $1.59 \rm{T_c} $ and $2.22 \rm{T_c}$, respectively, which
are found higher than the estimate in the absence of strong magnetic
field. Thus strong $B$ impedes the early dissolution of $Q \bar Q$ bound
states in the medium.