  We study the uniform approximation of the canonical conformal mapping, for a
Jordan domain onto the unit disk, by polynomials generated from the partial
sums of the Szeg\H{o} kernel expansion. These polynomials converge to the
conformal mapping uniformly on the closure of any Smirnov domain. We prove
estimates for the rate of such convergence on domains with piecewise analytic
boundaries, expressed through the smallest exterior angle at the boundary.
Furthermore, we show that the rate of approximation on compact subsets inside
the domain is essentially the square of that on the closure. Two standard
applications to the rate of decay for the contour orthogonal polynomials inside
the domain, and to the rate of locally uniform convergence of Fourier series
are also given.
