I know this is not the answer you want to hear, but there really is not too much that you are going to be able to do at home in this sitution. First of all - most broken bones in dogs will require surgery and it is only certain type of fractures in certain locations that can be treated with a splint. In a short-legged dog like a shi-tzu, this is especially true. Also, an improperly applied splint can actually interfere with the blood circulation and can do more damage than good. Rubberbands are likely to cause this to happen so I would strongly recommend you remove them immediately. If the splint is not properly applied, this dog can develop gangrene in the leg within a few days.
I know you said that you think the dog has a broken leg, but it is also possible that the leg is not broken but rather the dog has a torn ligament in the knee or something like that in which case a splint would not be helpful at all. Unless you know that this dog had a trauma that caued a broken leg, the torn ligament is a much more likely cause for lameness in the rear leg of the dog.
You stated that you had no access to a vet - can you explain that further? Is it a financial issue or possibly transportation???
If this is an financial issue, here is a website that lists several sources for financial aid for people with pets. I do not know how many of these currently have money available, but you can try contacting them to find out. You may also check to see if your local humane society is able to help you out in this situation. http://www.speakingforspot.com/helppayingforveterinarycare.html
If you truly cannot get this dog to a vet, then the best thing you can do is to keep this dog confined in a small space (like a cage or crate) for several weeks, taking him outside to go to the bathroom, but otherwise in the cage. I would not recommend trying to bandage or splint this leg unless/until you know if there is a fracture as an improperly applied splint could make the situation much worse than it currently is.
The only over-the-counter pain medication that can safely be used in dogs is aspirin. ASsuming about a 15 pound dog, this dog could have 1 baby aspirin twice daily to help with the pain. There are certainly more effective pain medications available for dogs, but they are by prescription only and will require a trip to the vet. NEVER give a dog tylenol or ibuprofen as these can be very toxic in dogs.
I hope this information is helpful. If so, please click Accept. If you have additional questions on this topic, please let me know.