I would think you would be pretty safe using beet spacings (16/sq.ft.). You want to pick turnips reasonably small anyway for best taste.

Okay well according to the book I have on square foot gardening, you're supposed to plant 9 beets per
1' x 1' square. However you did say that turnips should be picked when they're small for the best taste.

My question is then, since you recommended picking turnips
when they're small should I pick them when they're "baby turnips"? By this I mean should I harvest them when they're not fully mature? It takes 62-63 days for them to become fully mature; this includes the germination time
which is 7-10 days.

How many days should I allow them to grow for before I harvest them as baby turnips if I should harvest them
while they're small (a.k.a. immature). Will they continue
to mature after they're harvested/picked?

My wife and I pick them when they are 2" to 3" in diameter. Most turnip varieties will certainly grow larger, if you let them, but the taste and texture seems to decline rather substantially as they get really large.

At 3", planting 9/sq.ft. seems perfectly reasonable.

You can usually get a good feel for size by feeling around the plant. Don't focus too much on the days to maturity - when they are big enough, you can feel it.

jumping right in, Do you use turnips here for cooking, such as stew? they look like turnips I am used to (uk) but the seed catalogs talk about salad and pickling! which I am not used to, we do pickle them but as part of a mix, any help welcome, thanks.

The only way I've ever had turnips has been roasted. (Brought up British in the U.S.)

There are early turnips and there are late season turnips. The early are small the late ones are huge and take a lot of time and space to grow. 9/sq is good for early ones. Late ones you may crowd them at 4/sq even. The package description will be a good guide.

Here in the south we cook the greens along with the roots. Cook the greens as you would collard, or mustard, and dice up the root with it. You can also dice and boil them, drain and toss with butter, salt and pepper. Or boil and make a mash with some sausage or ham..
good luck and don't pay as much attention to the days as you might check the roots to see if they are big enough. Just sweep away a little soil, check, if not big recover. I'm not talking about digging up, just check the tip top of the root and u can get a feel of how big it is.

Here is a link that might be useful: cajunkorner

I successufully planted turnips at 9/square foot. But make sure to thin to just 9 plants when they come up. Otherwise the turnip won't have room to flesh out the root. And I agree with the others who advise to pick by size, not the number of days. The days is just a guideline. Environmental factors can cause things to go faster or slower. For me this is not quite enough sun, so everything takes longer in my garden.