Scattering of two mesons is considered in the framework of two-dimensional QCD in the large-$N_c$ limit with four different quark flavors. The scattering takes place through two coupled channels, corresponding to direct and quark-exchange processes. Infrared finiteness of the scattering amplitudes to order $1/{N_c}^2$ is shown. The theory reduces, at low energies, to an effective theory of mesons, interacting by a quark-exchange process, by means of a contact term. The unitarization of the scattering amplitudes leads to the emergence of a tetraquark bound state.
