Why do I run out of breath halfway through a phrase?

Usually not because you took too little air in, but because too much of it escapes at the start of the phrase. The measurable version of this is how long you can spin out one steady hiss on a single breath: twenty seconds is a reasonable target, and getting there is a matter of releasing the air evenly rather than of breathing in harder. Breathing low and wide rather than lifting the shoulders is the other half.