What is the function of the bulb in a Wien bridge oscillator?
An oscillator must have an "average" gain of exactly 1.000000 to produce stable oscillation.  If the gain is anything below unity, the oscillations will decay to nothing.  If the gain is anything above unity, the oscillations will grow until the oscillator can no longer manage unity gain (generally as a consequence of clipping parts of the signal).  For an oscillator to produce a distortion-free output, the gain must be uniform throughout each oscillation cycle.

One approach to designing an oscillator is to make it produce slightly lower gain at higher amplitudes.  If one has an oscillator circuit with a gain of 1.001 at voltages within the range -0.9 to +0.9, and 0.999 at values outside the range -1.1 to +1.1, then it will oscillate with an amplitude somewhere around a volt.  Unfortunately, the variation in gain throughout each cycle will cause some distortion.  One could reduce the distortion by reducing the variation in gain, but the circuit would become very "touchy"; even with the values given, something which reduces the gain of the circuit by 0.1% could cause it to stop oscillating altogether.

The idea of the light bulb is to have a circuit whose gain will be reduced as its output amplitude increases, and increased as its output amplitude decreases.  The light bulb's delay in reacting to changes in amplitude, and the delay in the circuit's amplitude in response to gain mean the light bulb/gain circuit may oscillate at a few hertz, but one can design that "oscillator" so that its oscillations will decay toward nothingness.