We report on experimental generation and evolution of circumstance-susceptible, narrow-bandwidth, h-shaped pulse in a thulium-doped fiber (TDF) laser. With typical mode-locking technique based on nonlinear amplifying loop mirror, a type of h-shaped pulse is generated in a net normal dispersion regime for the first time to our best knowledge. Different from pulses with similar profiles achieved in typical anomalous dispersion regime, the h-shaped pulse here exhibits extremely narrow spectral bandwidth and meanwhile becomes highly circumstance-susceptible. Not alike the well-preserved h-shaped profile with anomalous dispersion, here the h-shaped pulse can easily evolve into various other pulse patterns with circumstance variations, including peak-depressed profiles, burst-like emission, multiple h-shaped pulses, and even some highly complex temporal cases. Despite that, the h-shaped pulse broadens as the pump power increasing, being a typical pump-related characteristic dominated by the peak-power-clamping effect. Moreover, it is observed that the h-shaped pulse profile can be re-shaped by incorporating a piece of unpumped TDF into the cavity, i.e., introducing some reabsorption. Our results substantiate the experimental revelation of such a type of particular-profile pulse in the normal dispersion regime, demonstrating some new evolution features facilitated by the dispersion-relevant circumstance-susceptibility.