Large dynamic range current sense 1uA - 100mA
Referring to your second schematic, the output voltage of U1 is \$V_O=V_{R1}\cdot G + V_{REF}\$, and \$V_{REF}\$ is generated by U2.

Assuming U2 has some negative feedback, it works so that its negative input is at 0 V, so basically \$V_{REF} = -V_{O}\$.

Putting the two equations together you get \$V_O=\frac{V_{R1}\cdot G}{2}\$.

But is there any negative feedback? Let's try to "prove" it. Assume \$V_O = 0, V_{REF} = 0\$, this is a valid state for the circuit.

Some small variation happens at \$V_O\$, so it increases slightly. The negative input of U2 increases, the output decreases until the negative input is back to ground.

What if I redraw U2 circuit like this:







simulate this circuit &ndash; Schematic created using CircuitLab