Misconception
"In the Book of Mormon when Shiz is beheaded it says afterwards he 'struggled for breath.' Everyone knows you die instantly without your brain, so the Book of Mormon is wrong, and the church is wrong too."
Documented Truth
The verses in question are Ether 15: 30-31 which reads:
"And it came to pass that when Coriantumr had leaned upon his sword, that he rested a little, he smote off the head of Shiz.And it came to pass that after he had smitten off the head of Shiz, that Shiz raised up on his hands and fell; and after that he had struggled for breath, he died." (Ether 15: 30-31, The Book of Mormon)
This seems pretty cut and dry as a blatant falsehood within the text of the Book of Mormon. How can someone 'gasp for breath' with no head? However, if we look at the actual process of decapitation and how it affects the body, we learn some interesting facts.
Here is what science says about the physiology of decapitation:
"Decapitation is quickly fatal to humans and most animals. Unconsciousness occurs within seconds without circulating oxygenated blood (brain ischemia). Cell death and irreversible brain damage occur after 3–6 minutes with no oxygen, due to excitotoxicity. Some anecdotes suggest more extended persistence of human consciousness after decapitation, but most doctors consider this unlikely and consider such accounts to be misapprehensions of reflexive twitching rather than deliberate movement, since deprivation of oxygen must cause nearly immediate coma and death ("[Consciousness is] probably lost within 2–3 seconds, due to a rapid fall of intracranial perfusion of blood").A laboratory study testing for humane methods of euthanasia in awake animals used EEG monitoring to measure the time duration following decapitation for rats to become fully unconscious, unable to perceive distress and pain. It was estimated that this point was reached within 3–4 seconds, correlating closely with results found in other studies on rodents (2.7 seconds, and 3–6 seconds).The same study also suggested that the massive wave which can be recorded by EEG monitoring approximately one minute after decapitation ultimately reflects brain death. Other studies indicate that electrical activity in the brain has been demonstrated to persist for 13 to 14 seconds following decapitation (although it is disputed as to whether such activity implies that pain is perceived), and a 2010 study reported that decapitation of rats generated responses in EEG indices over a period of 10 seconds that have been linked to nociception across a number of different species of animals, including rats." (Wikipedia, Decapitation, https://en.wikipedia.org/wiki/Decapitation)
According to these studies "instant death," is really somewhere between 2-14 seconds. There are also many accounts of post mortem muscular "twitching" or "spasms" beyond that time frame that doctors say can be confused for conscious actions.
So what Ether describes during the death of Shiz is actually quite acurate for a decapitation. The body raising up on its hands as a last muscle spasm, and the head (most likely) gasping for breath when it suddenly has no connection to the lungs makes perfect sense. And all of this occuring within the 2-14 seconds after the death blow is completely feasible.