We investigated a sample of 15 luminous high-redshift quasars (3.3 z 5.1) to measure the mass of their supermassive black holes (SMBH) and compare, for the first time, results based on C IV, Mg II, and Hβ emission lines at high redshifts. Assuming gravitationally bound orbits as dominant broad-line region gas motion, we determine black hole masses in the range of M_bh 2 × 10⁸ up to M_bh 4 × 10¹⁰ M. While the black hole mass estimates based on C IV and Hβ agree well, Mg II typically indicates a factor of ~5 times lower SMBH masses. A flatter slope of the Hβ radius-luminosity relation, a possibly steeper slope of the Mg II radius-luminosity relation, and a slightly larger radius of the Mg II broad-line region than for Hβ could relax the discrepancy. In spite of these uncertainties, the C IV, Mg II, and Hβ emission lines consistently indicate supermassive black hole masses of several times 10⁹ M at redshifts up to z = 5.1. Assuming logarithmic growth by spherical accretion with a mass-to-energy conversion efficiency of = 0.1 and an Eddington ratio L_bol/L_edd calculated for each quasar individually, we estimate black hole growth times of the order of several ~100 Myr which are smaller than the age of the universe at the corresponding redshift. Assuming high-mass seed black holes (M = 10³-10⁵ M) the SMBHs in the z 3.5 quasars began to grow at redshifts z 4, while for the quasars with z 4.5 they started at z 6 to 10. These estimated timescales for forming SMBHs at high redshifts, together with previous studies indicating high quasar metallicities, suggest that the main SMBH growth phase occurs roughly contemporaneously with a period of violent and extensive star formation in protogalactic nuclei.