Hello Calire3,
You have not done anything wrong, 2218, and also 2219, 2220, and 2223 are not a true Turks Head because the end does not terminate at the begining of the knot. A true globe knot, one you can double, is more like 2216 and 2217 where the working end terminates at the same location that the standing end enters the knot.
If you would like to tie large globe knots, there is the Globe Knot Cookbook. Also, I have posted a few grids at the Knot Heads website (knww.net/Tutorials/Tutorials by Patrick Ducey) for globe knots. The grids are pretty easy to print and all will be capable of doubling the plies.
Good Luck!
Pat Ducey