unusual sink draining problem

[cam64uk]

when the kitchen sink is being used without the plug in, the water drains really slowly and backs up for a few seconds. But when I put the plug in, fill the sink up all the way to the top and take the plug out it drains really quickly, I have taken all the plastic pipes out underneath and cleaned them all. there was a bit of crud in the but not much. Put it all back together and the same problem exists. (I've also checked the overflow on the sink and that's clear. the house is relatively quite new but there is no warranty and to be honest, the plumbing quality is quite poor. we're also having similar issues with the showers. the whole system is connected to a Klargester Sewage Treatment Plant which appears to be working ok and the toilets are not affected. open to suggestions

 

[Deleted member 120897]

A photo of under sink waste-pipe arrangement would be useful.