The thicker the drywall, the less sound is transmitted. So if you installed 5/8" drywall over the 1/2" you would have a total of 1 1/8" thickness.
You could fill the stud cavities with blown in cellulose insulation.
When I built my house, I built the bathroom walls using 2X6 top and bottom plates and off-set 2X4 studs.
Then I wove insulation bats in and out before installing 1/2" drywall to close the walls in.
As sound is 'telegraphed' through the studs, using off-set studs provides a 'break' between the two drywall surfaces!
In your case, if space permits, you could build another insulated wall, making sure there is no contact between the two walls and then finish the new wall off with 5/8" drywall.