Mahler's problem asks for the largest possible value of the Mahler measure, normalized by the $L_2$ norm, of a polynomial with $\pm1$ coefficients and large degree. We establish a new record value in this problem exceeding $0.95$ by analyzing certain Turyn polynomials, which are defined by cyclically shifting the coefficients of a Fekete polynomial by a prescribed amount. It was recently established that the distribution of values over the unit circle of Fekete polynomials of large degree is effectively modeled by a particular random point process. We extend this analysis to the Turyn polynomials, and determine expressions for the asymptotic normalized Mahler measure of these polynomials, as well as for their normalized $L_q$ norms. We also describe a number of calculations on the corresponding random processes, which indicate that the Turyn polynomials where the shift is approximately $1/4$ of the length have Mahler measure exceeding $95\%$ of their $L_2$ norm. Further, we show that these asymptotic values are not disturbed by a small change to make polynomials having entirely $\pm1$ coefficients, which establishes the result on Mahler's problem. We also estimate that the limiting value of the normalized $L_1$ norm of these polynomials exceeds $0.977$, in connection with a question of Newman.