As a 16 year old man, would testosterone speed up my physical development during puberty?

No. Unless you have a deficiency, it shouldn't. And a true abscence would lead to non-formation of male genitalia. The presence of testosterone should be pretty clear with the deeper voice, facial hair, acne, sex drive. Let your body do it safely on its own. Low testosterone is typically diagnosed in older men, and people who previously used so many Anabolic steroids their body shut down production.
Be careful! Testosterone is one of the many hormones and chemicals that stimulate physical development during puberty. How these chemicals work together is very complex. Testosterone therapy is usually used only in patients who have some other disorder that prevents them from developing normally. Replacing testosterone can be very dangerous and have serious side effects if not done by an expert!