The conjecture that the safety factor profile, q(r), controls the improvement in tokamak plasmas from poor confinement in the Low (L-) mode regime to improved confinement in the supershot regime has been tested in two experiments on the Tokamak Fusion Test Reactor (TFTR). First, helium was puffed into the beam-heated phase of a supershot discharge which induced a degradation from supershot to L-mode confinement in about 100 msec, far less than the current relaxation time. The q and shear profiles measured by a motional Stark effect polarimeter showed little change during the confinement degradation. Second, rapid current ramps in supershot plasmas altered the q profile, but were observed not to change significantly the energy confinement. Thus, enhanced confinement in supershot plasmas is not due to a particular q profile which has enhanced stability or transport properties. The discharges making a continuous transition between supershot and L-mode confinement were also used to test the critical-electron-temperature-gradient transport model. It was found that this model could not reproduce the large changes in electron and ion temperature caused by the change in confinement.