How to prototype current sense circuit on bread board?
Breadboard prototypes are meant to verify the basic operation conditions of a circuit. They are not meant as a replacement of the "real thing". 

Also breadboards are only suited for prototyping simple circuits. Nobody would prototype e.g. a 3GHz circuit on them.
Use your breadboard to check if the results are nominally correct, then go to the real thing. 

In this case you make a reasonable 'current' sense circuit by keeping the sense resistor with the main current connectors outside the breadboard. (You can make them as big as you want). Then only connect the voltage sensing lines. 



Had troubles to remember the name ( I am getting old): Kelvin Circuit!