I don't grow carrots, but what I've seen says they don't take transplanting very well. Most people seem to plant them fairly thickly (because it's hard to manage the tiny seeds) and then thin them.

Thanks! I guess I'll try transplanting half the row of seedlings when they grow a little, to see what happens. This video seems promising: https://www.youtube.com/watch?v=35NcOLRkcPg

The reasoning I'd been told for avoiding transplanting them is that if you don't put them back in at exactly the angle they came up, you'll get crooked carrots.

There are good reasons not to transplant carrots: carrot fly, damage to tap root for example. However once you are aware of these issues and mitigate them there is no reason you can't transplant, though it is very time consuming and you may decide thinning is quicker. My process is to transplant when they are very small, only just showing the tip of real leaf growth. To scoop up the whole of the root area as much as possible (i use a chemistry spatula, plastering spatula, transplant scoop or tiny spoon), you need to dig down a couple of inches to loosen soil so you can ease the carrot out without damage to it or it's neighbours. Then i use a pencil to poke a hole to the same depth as the carrot root system, hold it at the right height in the whole and then gently scoop back soil in to fill. It is laborious and really only worthwhile if you have sown relatively thinly to begin with, otherwise the damage to neighbours and transplant may mean you lose the lot.