Give it two weeks and the "stitch" should go away with the same amount of exercise.
The side stitch you are experiencing is actually adaptive. It is the key to fitness. That side stitch is the result of inadequate oxygen to your kidneys (hypoxemia.) That hypoxemia triggers a variety of adaptive responses. 2,3-DPG will be released. This will change your red blood cell's affinity for oxygen making it easier for oxygen to be released. It will also result in the concentration of your blood (hemoconcentration) by causing you to shed fluid so your blood is thicker and carries more oxygen per unit of volume. (Your hemoglobin/hematocrit will increase.) Lastly, your kidneys will release erythropoietin (i.e. the hormone Lance Armstrong and others chemically doped with) that will cause your bone marrow to produce more red blood cells.
The main effects from hypoxemia will kick in about 7-10 days after you experience that side stitch and peak about 2 weeks after. That will be your moment of maximal fitness.
If you do not continue to stress your body at that point - your 2,3-DPG levels will drop and your body will slide back to its normal state.
This adaptive cycle is likely why all those recent studies you may have been seeing on high-intensity training (HIT) have shown it to be just as effective as long cardio sessions (i.e. training just 1 minute three times a week is just as effective as training one hour three times a week on FiO2 uptake and other measures of fitness.)
In sum, the "stitch" in your side is actually the key to achieving higher states of fitness. About two weeks after feeling a "stitch" will be your peak moment of adaptation from that hypoxemia causing that "stitch." If you can drive yourself to feel a "stitch" every time - you are taxing your cardiovascular system enough to trigger adaptations to higher levels of fitness.
"Fitness" is a high energy state. Though it increases adaptability to stresses and decreases short term mortality - there is a clear underlying metabolic cost that the body truly wants to avoid. Your body will continually try to slip back to that lower energy state and make it so the "stitch" returns.
Someday we may discover why the body seems to work so hard to be an "unfit" state.