Yeah, take your left thumb off of the back of the neck. Seriously. That way you can feel how you should be using your left hand. If your thumb is getting sore it means you probably have a death grip on the neck which is going to fatigue your hand, limit your mobility and eventually cause injury.
You should be able to stop the notes on the neck with your left hand by pulling the bass back against your belly, no thumb required. Now your fingers should also be right up against the fret. If you are playing in the middle of the fret or towards the back you have some bad geometry going on there and work a lot harder than you need to.

<- Bad

<- Bad Notice position of thumb and fretting finger

<- My suggestion for practice. See how it feels. Your thumb should just act as a guide.

<- Correct finger position for stopping a note.