2003 Dodge Stratus Car will start with a jump but dies when

Electrical problem
2003 Dodge Stratus 6 cyl Manual 97,000 miles

I had problems with my battery and recently replaced it. I am able to jump start the car, but the engine dies immediately after any jumper cable is disconnected. The "new" battery that I am now using only causes the headlights to come on right when connecting it and the icons on the dash don't appear normally (unless jump starting from another vehicle with "new" battery disconnected). Could I have damaged something from jumping from another vehicle? Could the new battery be defective? Why will the car start properly when jumper cables are connected but die immediately once the first cable is removed?

Thank you