I don't use a handlebar mount, but to keep my phone dry in a jersey pocket or a running belt, I just keep it in a 4 x 6 heavy ziploc bag. The bags do wear and start to develop pinholes after a few weeks of consistent use, so it's kind of wasteful, but an actual drybag or waterproof case just makes a phone too big to fit in a pocket or belt (and I'm maybe halfway through the 100 bags I bought four and a half years ago ).

I don't think you'd want to put a ziploc in a handlebar-mounted clamp for outdoor riding - you'd have to clamp it down tighter, and it would get more vibration, meaning it would probably wear pretty quickly - but I think for indoor use it would be fine.