We study the kernels $K_{n,s}(z)$ of the remainder term $R_{n,s}(f)$ of some Gauss--Turán--Kronrod quadrature rules for analytic functions when the weight function is the chosen subclass of Gori--Micchelli weight functions. We investigate the location on the elliptic contours where the modulus of the kernel attains its maximum value, which leads to effective error bounds of Gauss--Turán--Kronrod quadratures.