Can you get a new iPod if your iPods screen is broken under warranty?

My first suggestion would be to contact Apple Care. They will let you know first off if you are in your 90 days of phone support, if you are they will let you know if you are still in warranty. But a broken screen is considered a cosmetic damage, which is not supported by warranty. You can possibly "repair" it but it is normally cheaper to just buy a new one. If you didn't purchase the Apple Protection Plan, you will be charged extra for repair as well for shipping costs. You can request an appointment at an Apple Retail Store at Apple.com/retail or the Apple Care Support can make an appointment for you at a location close to you and even refer some of your information to them ahead of time.