4-Wire Strain Gauge Measurement

I want to accurately measure the resistance of a single strain gauge, unaffected by lead wire resistance and with ratiometric ADC conversion to compensate for excitation drifts.
I know for RTD measurements its common to use a simple circuit like below:

^{simulate this circuit – Schematic created using CircuitLab}

Unfortunately this works poorly for strain gauges, since the full swing of the signal is very low, on the order of V_exc/2 +- 10 mV, so a lot of the usable signal vanishes into ADC noise.

Naturally a bridge configuation + instrumentation amplifier comes to mind, but after trying for quite some time I just cannot come up with a decent 4-wire connection. Best I can find is a 3-wire one like this:

Since there a a lot of 4-wire strain gauge setups available commercially, there must be solution to this that I cannot find. Can someone point me in the right direction here?