This paper is concerned with approximation on variableLρp(·)spaces associated with a general exponent functionpand a general bounded Borel measureρon an open subsetΩofRd. We mainly consider approximation by Bernstein type linear operators. Under an assumption of log-Hölder continuity of the exponent functionp, we verify a conjecture raised previously about the uniform boundedness of Bernstein-Durrmeyer and Bernstein-Kantorovich operators on theLρp(·)space. Quantitative estimates for the approximation are provided for high orders of approximation by linear combinations of such positive linear operators. Motivating connections to classification and quantile regression problems in learning theory are also described.