Hey there! So I’m new to this so bare with me

i have an 05 tC
it wouldn’t start, had all the signs of a bad starter so I put a new one in.
started right up but had very low idle
drove 5 minutes and it died on me, but started right back up.
drove another 5 minutes, it died again but wouldn’t start. Had a buddy say it was the battery so I put a new battery in and still nothing.
Don’t think it’s the alternator because it should start and run for a little while even with a bad alternator when it has a new battery.
I checked all fuses and relays and they are good.
tried jumping it off to no prevail.
Need help please.