Why are my carrots and potatoes popping out of the ground
I believe that carrots and potatoes do need water to grow. However, the Minecraft Wiki is very unclear about this and there is no way to know for sure. The best thing to do would be to go to the Overworld, plow a dirt block, and try to plant it with and without water. If it does need water, it would not be possible to grow these crops in the Nether.
Potatoes and carrots do need water to grow. This is because you right click on a dirt block. To let it stay in that state it needs water. Its like irrigation.