Full Bridge Rectifier - Voltage Drop Under Load
I don't see anything unusual here.

You are trying to measure the dc voltage across a diode while giving the bridge an ac input. What you are actually measuring is some kind of average voltage across the diode, alternating between the forward and reverse bias cases.

If your transformer provides 5V rms then you would have a peak voltage of about 7V with no load. After the bridge rectifier a peak ac voltage with no load of about 5.5V would be reasonable. However, as soon as you start to draw current the average voltage, the dc voltage, will drop. That is simply the nature of the beast.
According to www.electronics-tutorials.ws the equivalent dc voltage from a full-wave rectified ac voltage is just 0.9 times the rms value.