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Fluvial planation surfaces, such as straths, commonly serve as recorders of climatic and tectonic changes and are formed by the lateral erosion of rivers, a process that remains poorly understood. Here we present a study of kilometer-wide, fluvially eroded, low-relief surfaces on rapidly uplifting folds in the foreland of the southwestern Tian Shan. A combination of field work, digital elevation model analysis, and dating of fluvial deposits reveals that despite an arid climate and rapid average rock-uplift rates of 1-3mm/yr, rivers cut extensive (>1-2km wide) surfaces with typical height variations of <6m over periods of >2-6kyr. The extent of this beveling varies in space and time, such that different beveling episodes affect individual structures. Between times of planation, beveled surfaces are abandoned, incised, and deformed across the folds. In a challenge to models that link strath cutting and abandonment primarily to changes in river incision rates, we demonstrate that lateral erosion rates of antecedent streams crossing the folds have to vary by more than 1 order of magnitude to explain the creation of beveled platforms in the past and their incision at the present day. These variations do not appear to covary with climate variability and might be caused by relatively small (much less than an order of magnitude) changes in sediment or water fluxes. It remains uncertain in which settings variations in lateral bedrock erosion rates predominate over changes in vertical erosion rates. Therefore, when studying fluvial planation and strath terraces, variability of both lateral and vertical erosion rates should be considered.