I'd stick with the gamebird feed, but chicks can have it too until they are close to laying eggs.
I wasn't fair to say that I feed an over the counter All Flock. We have a bulk mix with a 18% protein of ground corn and soybean meal with a poultry conditioner added. So we have all those good levels of vitamins and such.At 4 weeks old, turkey poults should still be on a high protein turkey or gamebird starter. Your all flock does not have the proper levels of protein, lysine, methionine or niacin they should be getting.
A high protein (28-30%) turkey or gamebird starter is recommended for the first 6 to 8 weeks followed by 6 weeks on a turkey or gamebird grower (24% protein) and then you can switch to a 20% protein all flock feed.