We perform a detailed analysis of broad pulses in bright gamma-ray bursts (GRBs) to understand the evolution of GRB broad pulses. Using the temporal and spectral properties, we test the high-latitude emission (HLE) scenario in the decaying phase of broad pulses. The HLE originates from the curvature effect of a relativistic spherical jet, where higher-latitude photons are delayed and softer than the observer's line-of-sight emission. The signature of HLE has not yet been identified undisputedly during the prompt emission of GRBs. The HLE theory predicts a specific relation, F_{nu},Ep_{propto}E_p_^2^, between the peak energy Ep in {nu}F{nu} spectra and the spectral flux F{nu} measured at Ep, F_{nu},Ep_. We search for evidence of this relation in 2157 GRBs detected by the Gamma-ray Burst Monitor on board the Fermi Gamma-ray Space Telescope from 2008 to 2017. After imposing unbiased selection criteria in order to minimize contamination in a signal by background and overlaps of pulses, we build a sample of 32 broad pulses in 32 GRBs. We perform a time-resolved spectral analysis on each of these 32 broad pulses and find that the evolution of 18 pulses (56%) is clearly consistent with the HLE relation. For the 18 broad pulses, the exponent {delta} in the relation of F_{nu},Ep_{propto}Ep^{delta}^ is distributed as a Gaussian function with a median and width of 1.99 and 0.34, respectively. This result provides a constraint on the emission radius of GRBs with the HLE signature.