iPhone keeps shutting off, works well when charging

Hi guys and thanks for your help.

My Iphone 5s keeps turnig off, but after I'm able to turn it on immediately as normal... and the phone works well, untill it shuts off again...

The problem more often occours when battery is low. When battery is more than, let's say, 50% it doesn't happen. When it's less than 15% and it turns off, I turn the phone on again only to see it shutting off after ios is started ( or after a couple of minutes it started).

While charging everything is perfect.

Any other function works well.

The battery charges well and the power consumption is high, but I think normal for an Iphone 5s, the phone lasts a little less than a day under normal/high use.

Sometimes battery percentage acts strange, it can go from 10% to 8% immediately but after it stays at 8% for sometime.

The funny thing is that I already changed the battery, but prior to this change, I was able to turn the phone on again only by connecting it to the charger...

It's possible that I have bought a defective battery???