Sue
I see 3 potential problems none of which are insurmountable.
Firstly, as RAH states, a tripod (or a bean bag ie some form of support) would help at high zooms. The image stabilisation is good but is not infallible and will struggle at high zoom particularly in less than bright light. Using the TCON adapter will make it worse.
Secondly I assume you are using fully programmable mode , if so I would recommend using aperture priority for long zooms so as to minimise shutter speed.
Thirdly and the most likely cause, there is little depth of field at max zoom and, say f2.8 aperture. When photographing birds with the FZ20 you have to set the autofocus to spot (and metering also to spot). Then practice in your garden or wherever taking bird shots at 6-10 metres preferably with trees/bushes behind. From my experience with the FZ20, this ain't easy at first as you have to hold the camera reasonably steady (tripod or bean bag again) for the A/F to fix on to the desired object. Other wise you will get a fuzzy bird with background leaves or branches being in focus.
I had a quite a few of the latter when first using the FZ20 so do not despair. I have had superb shots since once I mastered the spot auto-focussing
rgds david