Photoneutron cross sections were measured for 203 Tl and 205 Tl at energies between the oneand two-neutron thresholds using quasi-monochromatic γ-ray beams produced in laser Comptonscattering at the NewSUBARU synchrotron radiation facility. Our new measurement results in cross sections significantly different from the previously reported bremsstrahlung experiment, leading to rather different GDR parameters, in particular to lower GDR peak energies and higher peak cross sections. The photoneutron data are used to constrain the γ-ray strength function on the basis of the Hartree-Fock-Bogolyubov plus quasi-particle random phase approximation using the Gogny D1M interaction. Supplementing the experimentally constrained γ-ray strength function with the zerolimit E1 and M1 contributions for the de-excitation mode, we estimate the Maxwellian-averaged cross section for the s-process branching-point nucleus 204 Tl in the context of the 205 Pb -205 Tl chronometry.