The bulls broke into separate groups, and one of them trailed behind the rest, making the course extra unpredictable for runners trying to scamper out of their way.
Several daredevils slipped or tripped while the bulls charged down the narrow streets of the bull run route.
Television images showed one bull repeatedly tossing one runner in the air.
A 29-year-old Spaniard was gored in the knee and two other men were gored inside the northern city’s bullring at the end of the run, the regional government of Navarra said in a statement.
One of them was a 25-year-old from Florida who was gored in his leg and the other is a 29-year-old Spaniard who was gored in the groin, it added.
Three other runners were taken to hospital for injuries sustained in falls during the run.
The six bulls and six steers raced along the roughly 850-metre (2,800-foot) course from a holding pen to the city’s bull ring in three minutes and 12 seconds.
The bulls that run each morning are killed in afternoon bullfights by professional matadors.
The first goring of the festival so far this year happened on Saturday when a bull’s horn stabbed a 39-year-old Spaniard in the buttocks.
Monday was the fifth of the festival’s eight scheduled bull runs. They are followed by drinking, eating and concerts for the rest of the day.
The nationally televised early morning runs are the highlight of the nine-day festival made world famous by Ernest Hemingway’s 1926 novel “The Sun Also Rises”.
Officials called off the hugely popular event in 2020 and 2021 because of the Covid-19 pandemic, the first time the festival had been cancelled since Spain’s 1936-1939 civil war.
Sixteen people have died in the bull runs since 1910. The last death occurred in 2009 gored a 27-year-old Spaniard in the neck, heart and lungs.
His parents left a bouquet of flowers along the bull run route on Sunday on the 13th anniversary of his death.