Hiya!

I only made the carrot cake once so my advice is not that expert. But I can tell you what I did:
Actually I used a bigger cake tin than jamie says in his recipe. don't know the exact masses but it was a heart-shaped one and therefore neither very deep (more surface, less deepness so that it can bake through better).
Maybe you also have to make sure that when you grate the carrots they don't go watery and remain crunchy, so use a very sharp grate.
And last but not least: the cake is supposed to be very soft, so don't worry too much if it is not properly cooked in the inside.

hope that I could help!