Q.Does the bridge circuit produce any phase shift in the Wein bridge oscillator?

The Wien bridge oscillator is an RC oscillator whose frequency-selective feedback is a lead-lag network — a series $R_1C_1$ combination followed by a parallel $R_2C_2$ combination. At low frequencies the series capacitor $C_1$ acts as an open circuit and the output leads the input (lead network); at high frequencies the shunt capacitor $C_2$ acts as a short circuit and the output lags (lag network). Only at one particular frequency is the output a maximum with zero phase shift, and for equal components $R_1 = R_2 = R$, $C_1 = C_2 = C$,

This zero-phase, maximum-output property is what the oscillator exploits. A Wien bridge — the series $R_1C_1$ arm, the parallel $R_2C_2$ arm, and two resistor arms $R_3$ ($=R_f$) and $R_4$ ($=R_i$) — is connected between the op-amp's output and input. Since the op-amp is used as a non-inverting amplifier ($0^\circ$ shift), the feedback network must also give $0^\circ$; this occurs only at balance, so oscillation happens at the balance (resonant) frequency …