I have an idea: You don't have it wired correctly.

Get a voltmeter, measure the voltage on the power feeds (I assume there are two, every AM/FM head I've ever seen has two feeds, one is to be wired to battery, one is to be wired to switched power) when you turn the key from "off" to "accessory" then to "on".

I strongly suspect you will see the problem - it sounds to me like you have the "red" and "yellow" leads reversed. A common issue.
Most radios come with Red as "battery hot" and Yellow as "Ignition/accessory feed". Read the manual and heed the instructions about how to wire power to the new radio.