You either need to move the veggie beds further away from the privet, or remove the privet, if its yours. Privet does pinch a lot of nutrients, so growing near to a hedge of one isn't a good idea - and it must be delighted to have found such a rich source of nutrients to spread its roots into. Only other option I can think of is to clear out the beds, remove the soil, cut the privet roots back, and insert a root rhizome barrier vertically into the soil to a depth of 18 inches, then replace the soil from the beds to prevent the hedge roots reappearing.

That's 18 inches down from the original soil level.