Why won't my pictures transfer from my old iphone?

I got an iPhone 5s yesterday and I've been waiting for my pictures to transfer over all day and night. When I go to camera roll it says that there are the amount of pictures that I had on my old iphone ( a 4) but when I go into camera roll they're all just a bunch of gray squares and at the bottom it says downloading. Anyone know how to fix this? Btw I would rather not plug it into my compute because it's not a Mac and sometimes it screws up stuff, so if I could do this just on my phone that'd be great

You need to connect to iTunes so it's not that great.

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