You might want to give a Panasonic FZ200 a try. It's a superzoom with a relatively bright (constant f2.8) lens which you should be able to get for ~$600. Just make sure you can return it if it doesn't work out (because it might not because of the small sensor). And if it doesn't work out...
Unfortunately, I don't know of any bright (super)telephoto lenses for mirrorless cameras (Pany/Oly m4/3, Sony NEX, etc.) and I don't see that changing anytime soon since lenses that large will defeat the purpose of a small camera body.
So within your budget, that basically leaves us with an entry level DSLR (e.g. Nikon D3100 or D5100, Canon T3i or T4i, etc.) and third party (Sigma, Tamron, etc.) 70-200mm f2.8.
And I know you can't use a tripod, but how about a monopod? That would be immensely helpful since lenses like that can be quite heavy.
My granddaughter is a swimmer and I am tired of blurry pictures. I am ready to get a camera that will take swimming starts, action shots, but indoors so low lght also. I cannot use a tripod. I assume there is no pocketcamera that will do this. Does any of the newer cameras with interchangeable lenses work for this? I will get a DSLR if I have no other option. Need to spend less than $2000. Would like it as small as possible.