Tornadoes cause damage, injury, and death when intense winds impact structures. Quantifying the strength and extent of such winds is critical to characterizing tornado hazards. Ratings of intensity and size are based nearly entirely on postevent damage surveys [R. Edwards et al., Bull. Am. Meteorol. Soc. 94, 641–653 (2013)]. It has long been suspected that these suffer low bias [C. A. Doswell, D. W. Burgess, Mon. Weather Rev. 116, 495–501 (1988)]. Here, using mapping of low-level tornado winds in 120 tornadoes, we prove that supercell tornadoes are typically much stronger and wider than damage surveys indicate. Our results permit an accurate assessment of the distribution of tornado intensities and sizes and tornado wind hazards, based on actual wind-speed observations, and meaningful comparisons of the distribution of tornado intensities and sizes with theoretical predictions. We analyze data from Doppler On Wheels (DOW) radar measurements of 120 tornadoes at the time of peak measured intensity. In striking contrast to conventional damage-based climatologies, median tornado peak wind speeds are ∼60 m⋅s−1, capable of causing significant, Enhanced Fujita Scale (EF)-2 to -3, damage, and 20% are capable of the most intense EF-4/EF-5 damage. National Weather Service (NWS) EF/wind speed ratings are 1.2 to 1.5 categories (∼20 m⋅s−1) lower than DOW observations for tornadoes documented by both the NWS and DOWs. Median tornado diameter is 250 to 500 m, with 10 to 15% >1 km. Wind engineering tornado-hazard-model predictions and building wind resistance standards may require upward adjustment due to the increased wind-damage risk documented here.