To try answering the specific question:
Are there exercises that explicitly help to prevent side stitches to occur?
The deep and controlled breathing in on the right foot and out on the left foot is suggested to help. You have to be concentrated on this and train this while exercising to breath the right way without concentrating to much on this while performing.
According to the internal organs being tossed around, stretched and hammered or what else happens: Remember to always warm up properly and effectively before performing AND before exercising.
Strengthening the muscles in that area like the diaphragm as well as the core muscles like the lower back, the abdominals and the obliques is of course a direct treatment. Especially the core muscles improve your static and "bearing" parts, roughly spoken.
If you gradually increase exercise intensity the body will "warm up" while at it and will stretch out what hasn't already been.
Remember this during warm up, that you have to let the body experience the amount of pressure, you intend to put on it while performing, as well as getting your pulse as high, as you expect to get it - just over shorter periods.
See more here.
Does it get better, when I continue working out?
It is usually said just to continue will make it disappear. Especially by the coach. But as long as no real threat has been proven or documented (by some accident or injury actually happening) you might just keep up the good mood and use pure willpower to overcome the hard times.
It works for me to think about breathing with my belly instead of just the upper chest. It might touch some right places, but nevertheless it will activate the side stitch area and is worth a try.
Is it advisable to go to a doctor to check it out?
Well no, as there are no proven treatments and no proven actual danger. As far as to this days medical science, just continue running and exercising. It might disappear some day.
Another article with mere to read.