The puppy will consume 60*2 = <<60*2=120>>120 ounces of the special dog food in the first 60 days.
This leaves 365 - 60 = 305 days where the puppy will consume 4 ounces of food per day, or 305*4 = 1220 ounces of the special food.
Thus, the puppy requires a total of 120 + 1220 = <<120+1220=1340>>1340 ounces of food before it is 1 year old.
Each 5-pound bag contains 5*16 = <<5*16=80>>80 ounces of special dog food.
Therefore, the puppy will need to eat 1340/80 = 16.75 bags of the special dog food, meaning the family will need to buy 17 bags.
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