Based on daily observation records at 277 meteorological stations on the Tibetan Plateau (TP) and its surrounding areas during 1970–2017, drought evolution was investigated using the Standardized Precipitation Evapotranspiration Index (SPEI). First, the spatiotemporal changes in the growing season of SPEI (SPEIgs) were re-examined using the Mann–Kendall and Sen’s slope approach—the piecewise linear regression and intensity analysis approach. Then, the persistence of the SPEIgs trend was predicted by the Hurst exponent. The results showed that the SPEIgs on the TP exhibited a significant increasing trend at the rate of 0.10 decade−1 (p < 0.05) and that there is no significant trend shift in SPEIgs (p = 0.37), indicating that the TP tended to undergo continuous wetting during 1970–2017. In contrast, the areas surrounding the TP underwent a significant trend shift from an increase to a decrease in SPEIgs around 1984 (p < 0.05), resulting in a weak decreasing trend overall. Spatially, most of the stations on the TP were characterized by an increasing trend in SPEIgs, except those on the Eastern fringe of TP. The rate of drought/wet changes was relatively fast during the 1970s and 1980s, and gradually slowed afterward on the TP. Finally, the consistent increasing trend and decreasing trend of SPEIgs on the TP and the area East of the TP were predicted to continue in the future, respectively. Our results highlight that the TP experienced a significant continuous wetting trend in the growing season during 1970–2017, and this trend is likely to continue.