solution to guttering overflowing into the eaves

hi I was hoping people could help. My gutter overflows when it rains heavily the issue is that water then spills into the eaves onto the soffit which causes mould and in one extreme case it came into the ceiling.
The issue I believe is I have one of those gutters where the outside face is higher than the inside (which is possibly bent over the fascia) it has weep holes but in a heavy rain storm the water is still backing up over into the eaves. The downpipes simply can't get enough water away.
I have no tree or leave issues and I also recently had the stormwater flushed out at high pressure so it is definitely free flowing. I don't want to go to the extent of installing larger stormwater pipes underground (currently 90mm)
A plumber on the phone mentioned the solution of something that can be attached to the downpipe which when the stormwater is backing up which I think is the issue in a big storm then the water kind of flows out of a diverter which stops the stormwater overflowing in the guttering. Has anyone had any knowledge of this sort of thing and whether or not its effective?
I need a solution because I will be renting the place outand I want things set up so there's not damage occurring over the long run.
 
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