Enthalpy relaxations of propylene glycol and glycerol were followed in the time domain at 160 and 182 K, respectively, in the glass-transition region with use of an adiabatic calorimeter. The relaxation processes were analyzed and fitted with a stretched-exponential function with nonexponentiality parameter \ensuremath{\beta}. Specification of the temperature jump prior to the relaxation measurement, (\ensuremath{\Delta}T${)}_{\mathit{j}}$, was found to be indispensable for the characterization of the relaxation processes in the nonlinear-response regime and the value of \ensuremath{\beta} was studied as a function of (\ensuremath{\Delta}T${)}_{\mathit{j}}$. The value of \ensuremath{\beta} depends on both the sign and magnitude of (\ensuremath{\Delta}T${)}_{\mathit{j}}$, indicating a characteristic difference between the rates of creation and annihilation of structurally ordered regions in liquids. However, the values of \ensuremath{\beta} extrapolated from the positive and negative sides of (\ensuremath{\Delta}T${)}_{\mathit{j}}$ to zero degree Kelvin agree with each other (0.62\ifmmode\pm\else\textpm\fi{}0.02 for propylene glycol and 0.65\ifmmode\pm\else\textpm\fi{}0.02 for glycerol, respectively), and are in excellent agreement with results obtained by heat-capacity spectroscopy. This is an experimental verification that the enthalpy-relaxational observation in time domain is equivalent to that in frequency domain only for (\ensuremath{\Delta}T${)}_{\mathit{j}}$=0 K.