  We studied the field dependent thermal conductivity ($\kappa$) of
Na$_2$Co$_2$TeO$_6$, a compound considered as the manifestation of the Kitaev
model based on the high-spin $d^7$ Co$^{2+}$ ions. We found that in-plane
magnetic fields beyond a critical value $B_c \approx$~10 T are able to
drastically enhance $\kappa$ at low temperatures, resulting in a double-peak
structure of $\kappa(T)$ that closely resembles the behavior of
$\alpha$-RuCl$_3$. This result suggests that heat transport in
Na$_2$Co$_2$TeO$_6$ is primarily phononic, and it is strongly affected by
scattering from magnetic excitations that are highly tunable by external
fields. Interestingly, for magnetic fields $B // a$ (i.e., along the zigzag
direction of the Co-Co bonds), there is an extended field range which separates
the long-range magnetic order for $B\leq B_c\approx10$ T and the partially
spin-polarized gapped high-field phase for $B\gtrsim 12$ T. The low-energy
phonon scattering is particularly strong in this field range, consistent with
the notion that the system becomes a quantum spin liquid with prominent spin
fluctuations down to energies of no more than 2 meV.
