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Let μ denote a symmetric probability measure on [−1,1] and let (p_n) be the corresponding orthogonal polynomials normalized such that p_n(1)=1. We prove that the normalized Turán determinant Δ_n(x)/(1−x²), where Δ_n=p_n²−p_n−1p_n+1, is a Turán determinant of order n−1 for orthogonal polynomials with respect to (1−x²)dμ(x). We use this to prove lower and upper bounds for the normalized Turán determinant in the interval −1<x<1.