Need to Amplify 13mv to 5volts using a LM324 op Amp
By the first look it looks like you are sourcing to much current out of your opamp:

I = 5V / 47Ω ≈ 100mA


Gives you 100mA maximal output current but the LM324 is only capable of delivering 20 - 40mA.

Try with bigger values like:

R3 = 122kΩ
R4 = 47kΩ




EDIT:
Ok your resistor values are wrong too. The gain of your circuit is:

gain = (R4 / R3) + 1 = 1.4


And not 384.

EDIT2:

As others have pointed out you can't expect 5V on your output because the LM324 is not capable of driving the outputs close to its power rails.