I think that's a fair and honest assessment but stay with my for a moment...
Sensor size and lens zoom range and quality could well come into this. Some bridge cameras have the same sized sensor as a the small sensor compact but they could still off you better handling and possibly a better lens too but in a larger package. In your place I think I'd start by taking a look at 1" sensor bridge cameras as the image quality is likely to be a step up from the small sensor compacts and bridge cameras. If you find the prices too high then maybe look at a smaller sensor bridge.
For 1" sensor bridge cameras Panasonic and Sony make some of the front runners but there may be others I don't know about. There are a couple of threads about them on this site and maybe you can search for them or even better some of the contributors may provide links.
One thing I would advise you to do, well two things actually, read the manual and read up of the exposure triangle of aperture, shutter speed and ISO and perhaps tag depth of field and perspective onto the list of things to read up on.
Good luck choosing and I hope you'll come back and show us some of those bird and bunny pictures