How to find bridge current in this circuit?
You forgot to include one of these equations (that you already gave):
$$I_b = \frac{V_1^{out}-V_3^{out}}{R}$$
or
$$I_b = \frac{V_2^{out}-V_4^{out}}{R}$$
This should give you 5 equations and 5 unknowns.
[edit] I did not redo your calculations. Don't hold it against me if there was an error in there somewhere.