Assuming that the galvanometer has a coil in it then the effect is due to the relatively low total circuit resistance.
What this means is that we are dealing with a spring-mass system which is massively over-damped.
The mass is the coil and former and the spring is the return spring(s) which moves the coil to the equilibrium position when there is no current passing through it.
If the galvanometer is of a conventional type then the coil is probably wound on an aluminium former and the system is normally critically damped to enable the pointer to reach its final position in the shortest possible time.
The damping being achieved by an emf induced (Faraday) in the former because it is moving through a magnetic field which in turn induced currents in the former which oppose the motion of the former (Lenz).
If it is a ballistic galvanometer then the coil former will either be wooden or plastic and in normal use underdamped.
So what happens when a solenoid is connected to a galvanometer?
In effect there is coil and a complete circuit with a very low resistance.
As the coil moves through the magnetic field an emf is induced (Faraday) and because the resistance of the circuit of the circuit is so low the induced current is larger which produces a large opposition to the motion of the coil (Lenz) thus the system is heavily over-damped and the return to zero is slower than expected.
I suggest as an experiment adding a resistance box into the galvanometer and solenoid series circuit and noting the effect of changing the resistance which has been added. I would expect that depending on the value of the added resistance the maximum deflection might actually increase compared with having no extra resistance in the circuit and the return to zero might also be observed to be quicker.