Turkey frying oil temperature issues

If you are saying you cannot get the oil hot enough during pre-heating:

1. You may need a bigger burner than the one you are using. Most resources I've seen suggest over 100k BTU
2. There may be something physically wrong with your setup (i.e., the vessel should be closer to the flame)
3. The ambient temperature at the time of cooking was simply too cold for the burner to keep up (see problem 1).

If you are using a purpose-built commercial turkey fryer, I suggest consulting the book or contacting the company. If you built the rig yourself, I suggest more fire! (and lots of safety precautions)

Now... if you are talking about a temperature drop after you dunk the bird, realize that is totally normal.

Alton Brown suggests a preheat temperature of 250 degrees. He suggests raising the temp to 350 after the bird is in the bath but adds the following advice:

"Closely monitor the oil temperature: it's will start to climb above 350 as moisture cooks out of the turkey; reduce the gas flow accordingly..." (Ref ep. 163 transcript in his book Good Eats 2, page 413)

You may need to use a smaller turkey to achieve the internal temperature you want without burning the outside. I'd look for something in the ballpark of 10 pounds. If you are cooking for a crowd, it's best to go with multiple birds instead of going bigger.

Be mindful of carry-over cooking:

Even after it’s been removed from a heat source, food continues to cook. Its internal temperature can rise anywhere from 5 to 20 degrees. The larger the food, the more carryover or residual cooking you’ll have. http://www.cookingclarified.com/2011/01/carryover-cooking/

Does the hose on your turkey fryer have a safety valve? If so, this valve may be malfunctioning. The safety valve on my turkey fryer is battery powered and is set up to turn off the gas after 10 minutes unless you press the button to start the timer again. I have been frying a turkey for Thanksgiving for about 6 years and never had this problem with temperature. Until this year. The oil took much longer to get to 350 degrees and when the turkey went in it never got back up to that temperature. It turned out that the valve, although open, wasn't letting enough gas in.

I learned this the hard way by replacing it after Thanksgiving.

You also want to get the oil to 350 degrees before you add the turkey. I am puzzled by why the Alton Brown recipe suggests adding the turkey before it reaches the critical temperature; that advice is contrary to my own experience and any recipe that I have read

I’ve been doing this for years. I suggest getting the temp up to 375 before lowering the bird in, as the oil temp will drop significantly immediately afterward.

If it is too cold out, find a way to build an impromptu shelter to retain some of the heat and block the wind.

You are actually on the right track. Put the bird in at 250 and the oil will get hotter but never reach 350.
Alton got this one wrong. Start timing when the turkey starts vigorously frying and roiling bubbles. 3-3.5 minutes per pound and take it out and temp it. It may be up to 165 white/ 200 dark but will still be moist and tender.

You will even be happier if you inject the breast with broth or stock but not a whole lot of fancy spices.