Privets are very resilient. If it is Privet (I can't quite tell, and I'm no plant expert), you can hack it way down and it will bounce right back. But you should probably wait until late winter/early spring. Someone else can confirm, but I don't think this is the time of year for heavy pruning for this sort of thing.

If it makes you feel more confident about it...
Earlier this year (Feb/March) I cut my Privets WAY back. I'm talking like 20' tall bushes/trees (they were very overgrown) that we hacked back to about 4-5' of mostly just trunks/branches. They have already grown back to looking normal and bushy again. I was afraid I might kill them by cutting them back too far, but in fact I think they need to be cut down even lower next year.

you would need a close up pic of leaves.. if you want an ID....

just to make sure ... you get good info ...

ken

It wouldn't let me upload more than one photo, but I guess I can put a second one in a reply!

Oh yeah, that's a privet. A couple of years ago I cut some of mine down from @18 feet to @ 8 inches, and they have come back just fine (I am keeping them MUCH shorter). For what it's worth, I had to do my butchery in October, before the snowiest winter I'd ever seen, and the privets didn't miss a beat. Yes, the best time to cut them down is early spring, but they are basically cast iron weeds, so do what you have to when you have to. Be ruthless.