We study the effective bosonic string that describes confining flux tubes in three-dimensional $\mathrm{SU(N)}$ Yang--Mills theories. Although the low-energy properties are universal and well described by the Nambu--Gotō action, the subtle dependence on the gauge group is embedded in a series of corrections, which remain undetermined, appearing in the expansion around the limit of an infinitely long string. We extract the first two of these corrections from a set of high-precision Monte Carlo simulations of Polyakov loop correlators at finite temperatures close to the deconfinement transition. We present and compare the results of new lattice simulations for theories with $N=3$ and $N=6$ color charges, along with an improved estimate for the $N=2$ case, discussing the approach to the large-$N$ limit. We show that our results are compatible with analytical bounds derived from the S-matrix bootstrap approach. Additionally, we present a new test of the Svetitsky--Yaffe conjecture for the $\mathrm{SU(3)}$ theory in three dimensions, showing that our results for the correlator of Polyakov loops perfectly agree with the predictions obtained using a conformal perturbation approach to the two-dimensional three-state Potts model.