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I have a shrub that came with this house, and it is about 7ft tall and 3ft wide. It grows white flowers at the summers-fall, and is drought-tolerant. It grows in North Carolina, United States. It has small 1/2 inch flowers that are maybe white or yellow. The flowers come in the fall, are fragrant and don't produce berries.
This looks to me like Osmanthus fragrans , sweet/tea olive bush. https://hgic.clemson.edu/factsheet/tea-olive/
Looks like a Ligustrum variety, commonly known as privet, possibly japanese privet, Ligustrum japonicum. These bear fragrant (some people would say unpleasant smelling) white sprays/panicles of small flowers in late summer or fall, sometimes followed by berries, so it rather depends on whether that flower description fits the flowers you see. https://www.rhs.org.uk/plants/77548/Ligustrum-japonicum/Details
There are other varieties of privet, but japanese privet tends to have slightly thicker, waxier leaves than the other varieties, like the leaves in your photo appear to have.
Looks like a rhododendron to me, they grow all over NC. If the leaves are thicker/stiffer/tougher than most tree leaves I would be almost positive. To me, the leaves feel fake, like they are polyester.
It's a great plant, one of my favorites. When I was backpacking the Smokies they grew ubiquitously in the valleys. Enjoy it and don't be afraid to trim it or beat it into submission; it can take it, and if left unchecked could get too big for that space.
