[gaidzag]

You are electrowinnowing copper out of a saturated copper sulfate solution, and bubbles start forming at the cathode in addition to copper after you crank up the voltage.   Is that because the copper ions are being reduced quicker than new copper ions can migrate to take their place, causing water molecules to get cracked instead?  Is this why mechanical stirrers are used in plating baths?

[Enthalpy]

I'd suppose that, as you increase the voltage, more ions present in the electrolyte can react besides copper ions.

I know at least one different reason for stirrers. Electroplating is naturally an unstable process that deposits metal irregularly, because is a spike begins to form, it concentrates the electric field, and deposition accelerates there, which grows the spike faster. Heat acts against this natural instability by redistributing the ions that the concentrated field attracts to the spike, but only if the electrolyte is warm enough and at small current density. Stirring is an answer to that, and also to the depletion of the electrolyte at places where the target part is by design closer to the anode(s).