Getting shots from roughly 30 feet away

I'm having trouble getting shots from a distance. My camera goes up to 4x zoom (16x digital.) When I get around 20 feet from these birds I've been trying to capture at my feeder (Finches I Think), they fly away too quick. I don't have a mono/tripod, but I'm working on it. I tryed taking a few pics but they were quite blurry and I'm wondering how you guys would achieve this

You'll have to sit out there enough times to let them get use to you. And find a rest for the camera. Maybe sit in a chair and put your knees up to rest the camera on. Or, stay further back and crop the image to make it look bigger.