Magnetization measurements were carried out on bulk Tl2Ba2CaCu2O8 (referred to as TI-2212) and on various site-selective substituted Tl-2212 samples. At 5K between 0 and 4.5 T, the 5 at. % Mg-doped Tl-2212 (Tl,Mg-2212) samples displayed enhanced pinning as demonstrated by a field-dependent increase of the magnetic critical-current density J(c) by 18 to 25 percent over that of pristine Tl-2212. Excess Mg (10-15 at. %), however, is deleterious. Rietveld refinement of the x-ray diffraction pattern showed Mg on the Tl sites. Auger electron spectroscopy analysis showed part of the Mg on grain boundaries. The flux-creep activation energies are higher for flux expulsion than for flux penetration in both TI-2212 and Tl,Mg-2212 samples; the latter displays higher individual values. Our results demonstrate an increase in the number density of flux lines as a result of increased density of atomic-size-structural defects by Mg (5 at. %) doping. In the Tl2-yBa2(Ca1-zYz)Cu2O8-x. (z = 0-0.3; single phase; x and y represent oxygen and thallium vacancies) system also studied, the T(c) decreases as z increases. At z = 0.3, the sample becomes an antiferromagnetic semiconductor.