Be realistic about...
The simplest way to get the balance right is to buy a smaller bag or briefcase, and to promise never, ever, to supplement it with a Tesco bag.
To make this approach successful, you will have to make sacrifices. Begin by taking time at the end of each day to catch up with marking - don't take it home unless there is no other way to get it done. When you do leave school, you will feel you've achieved something and you will be less physically burdened - what do 35 Year 7 exercise books actually weigh?
Empty your bag before you leave work, and put back into it only the books and papers you will need for preparing tomorrow's lessons. This daily purgation will also help you avoid distressing archaeological discoveries as the term advances - finding forgotten but crucial papers when it's too late to do anything about them, or when they're crumpled beyond recognition.
Emotional baggage counts, too. Make your last thoughts as you leave school positive, and go through the door thinking about something that went well.
Dump the rest.