Toilets, Sinks, Showers, Tubs and Disposals - Steam Shower Water Supply

I have purchased a steam shower whirlpool combo untit from EBay (Chinese Manufacturer of some sort) Directions and manual are very poor. I was able to connect everything and all seems to work well save for one thing -when I turn the shower to cold water, the water supply is very very light (almost shuts off completely or sporatically squirts out of faucet). Cold water supply is split off (with a T joint) to run to cold water inlet supply and to steam generator unit. Hot water supply is not T'ed off and is connected to hot water inlet. ...Is there a way to get the cold water to come out better through the faucet? I would think that the cold water going into the steam generator is causing this problem but how do I correct it the right way?

Also not sure why there would be a problem to begin with. If not utilizing the steam generator, cold water supply should be same as hot water supply -however this is not the case. Any advice is much appreciated.

There is a monitor that shows on the control display unit if when running the steam shower the water supply is not sufficient. This monitor has never gone off however so I'm sure the steam shower is getting the water supply it needs.

either there is a problem in the faucet or the supply piping. which, i can not say from here but i would turn the water off to the faucet and dismantle the cold side and check for debris as a first step. maybe check the steam generator for flow, not threw it but to it.

You could put a valve in the line after the tee going to the steam generator, and see if all works well. If yes, you know where look, the steam generator. If still having problems, Check your supply line feeding the faucet, then check the faucet parts if need be.