If the belt is the correct size (width) for the pulleys and you tightened it properly, the belt cannot be the cause of the problem. Start the engine and check to see if the fan and hence the water pump is spinning. If the fan is not moving or barely spinning, then I suppose the water pump bearings are locking up or are locked up. The resultant slippage of the belt on the pulley would have caused your initial belt break.
If it all looks normal after visual inspection, then use an infrared thermometer and compare radiator inlet temp vs outlet temp. If you get a large variance (not sure what, but maybe more than 20-25 degrees or so) then I would start suspecting water flow issues. Or, even air flow across radiator.
