I'm allergic to beef in the same way that lactose intolerant people are allergic to milk. There are times at home when I cook ground turkey and I react the same way as when I eat beef, so I'm suspicious that either there's beef in the meat, or that they prepare ground beef at the same facility and they don't clean the machinery well enough. Is there at way to test for beef content at home?
Molecular biologist here - there's no simple or cheap way to do this at home. You could try approaching your local university and see if they would be interested in testing samples for you - food adulteration is a big issue.
There's also the possibility that you are allergic to the turkey as well. The American College for Allergy, Asthma and Immunology suggest that this is entirely possible. If you have one meat allergy you are more likely to have another. Having said that, if what you have is actually an alpha-gal allergy (e.g. as a result of a Lone-star tick bite), these are only found in mammalian meat, birds (like turkey) have no alpha-gal, so a reaction to turkey would likely indicate cross-contamination.
Do you know if you have an allergy or a food intolerance? If you know exactly what your body is reacting to there may be a way to test for it at home, but I'd suggest either finding a brand that you don't react to and stick to it or, better yet, grind your own at home as Joe suggested above.
It's pretty easy to grind your own meat. It also has the advantage that you can can make it exactly the way you like (fat percentage, parts used, etc.). You'll know exactly what's in it, and you'll be much less likely to get something you don't want, which could be offal or something even more awful.
How much horse would you like in your lasagne?
BBC News report from 2013 - Findus beef lasagne contained up to 100% horsemeat, FSA says
Even a 'trusted' source can have unexpected ingredients, if not correctly policed.
I second the suggestion to grind your own.