Christina Norstein was hoarse, her nose swollen by mask marks. She wasn’t sleeping much, and woke up crying one morning after a day at work. “I feel like we’re being sent to slaughter,” she said.

Ms. Norstein is an intensive-care nurse at Montefiore Medical Center’s Moses campus in the Bronx, one of the hospitals inundated with Covid-19 patients.

In hospitals across New York City and elsewhere in the country, nurses and doctors are complaining about a lack of safety equipment, insufficient staffing, murky policies and other challenges. New York City accounts for the largest number of Covid-19 cases in the U.S. On Tuesday morning, the city reported 40,900 cases and 932 deaths.

In the last few weeks, the 57-year-old Ms. Norstein and other nurses say they have seen freezer trucks out back for dead bodies; four to five patients dying every emergency-room shift; the loudspeaker frequently booming out “codes” for patients whose hearts or breathing stopped. Colleagues who were healthy one day fell critically ill the next.

Despite multiple alarms raised by workers in the past two months, she and other nurses say the hospital system stumbled in creating protocols to prevent the disease’s spread and is sending its workers out to the battlefield underprepared. Staffers say the hospital is running out of critical supplies—ventilators, dialysis machines, sedation medication, feeding pumps. Ms. Norstein said a doctor told her that physicians are discussing how they would decide who lives as ventilator supplies dwindle.