Phone dies prematurely (20%, 59%, 80%, 3%) unless plugged into charger

My iPhone 4 has been dying all the time. At various percentages - 20%, 60%, 4%, 80% - it doesn't seem to matter.

It happens when I am on various apps - email, text, camera, maps - and when I am on a phone call.

It tends to turn back on straight away when I press the power button, but can take quite a long time to load up.

It does not die if it is plugged into the wall charger, my laptop to charge, or the car charger. I am now only making calls when it is plugged in.

Is it simply that the battery is old and needs to be replaced (I bought the phone in 2010)? Or could it be a software bug?

Thank you, Maddie

Apologies if a repeat - I tried to find this Qn but couldn't.