Abstract We present the discovery from the TESS mission of two giant planets transiting M-dwarf stars: TOI 4201 b and TOI 5344 b. We also provide precise radial velocity measurements and updated system parameters for three other M dwarfs with transiting giant planets: TOI 519, TOI 3629, and TOI 3714. We measure planetary masses of 0.525 ± 0.064 M _J, 0.243 ± 0.020 M _J, 0.689 ± 0.030 M _J, 2.57 ± 0.15 M _J, and 0.412±0.040 M _J for TOI 519 b, TOI 3629 b, TOI 3714 b, TOI 4201 b, and TOI 5344 b, respectively. The corresponding stellar masses are 0.372 ± 0.018 M _☉, 0.635 ± 0.032 M _☉, 0.522 ± 0.028 M _☉, 0.626 ± 0.033 M _☉, and 0.612 ± 0.034 M _☉. All five hosts have supersolar metallicities, providing further support for recent findings that, like for solar-type stars, close-in giant planets are preferentially found around metal-rich M-dwarf host stars. Finally, we describe a procedure for accounting for systematic errors in stellar evolution models when those models are included directly in fitting a transiting planet system.