Shin splints is a bit of a lazy diagnosis by your GP - it's like diagnosing "headache". There are three primary causes of shin pain, the most likely of which is down to stiffness in the muscle that runs up the front of the shin - the tibialis anterior. This can be addressed by some stretching work - pointing your toes stretches the muscle - after you run.
You need to look into the cause of the pain. This is usually from either running too far, too soon; running on hard surfaces; or incorrect shoes.
If you haven't already done it, please go to a specialist running shop and have your gait analysed. The staff will record you running on a treadmill in neutral trainers and slow it down to see how your foot rolls when you run, they can then advise on the level of support needed in your shoes to help avoid injury. Your shoes will have a life of around 500 miles, so if you've run that far in them it is time to get new ones.
If you're exclusively running on pavements this can sometimes cause shin pain. If you can, try running on softer surfaces such as trails and see if that stops the issue.
The other primary cause of shin pain is building up your mileage too quickly. It is advised that you only increase your mileage by 10% per week to avoid injury. If you've jumped up by too much and started suffering pain then you may need to drop back on your mileage and build up more slowly.
It might be worth seeing a sports physio or sports massage therapist to get them to have a look and rule out any other problems such as stress fractures or compartment syndrome (also causes of shin splints but much less common).