When starting out, we so commonly jump straight into running. This is natural, perhaps, since we all remember running around so effortlessly when we were young children. But it is so easy to forget that we also skipped, jumped, climbed trees, and suspended and swung ourselves on the monkey bars, too. Few of us can do these things without risk as we get older, unless we have built up to them. The reason is simple: if we do not maintain vigorous or strenuous physical activity continuously throughout our lives, our size outgrows our physical capacity.
So even though you may not be physically heavy relative to the average, you may still be heavy in relation to your current physical strength. That is, you may have poor relative muscle, tendon, and ligament (joint) strength. And this is normal.
Contrary to the way we commonly think of it, running is very physically demanding. Peak ground reaction forces are typically 2–3 times body weight, while peak forces on the tibia (lower leg) are typically 6–14 times body weight! It is natural that you would feel ‘heavy’ when you run, if you have not been running consistently for a significant period of time.
So what can you do about it?
The obvious answer is to build your running volume up slowly until such time that it begins to feel more comfortable and manageable. But I would like to suggest something that is rarely considered.
In contrast to how we commonly view it, running is a moderately-advanced exercise. As such, it is often better to begin with a foundation exercise—in this case, walking. Walking conditions all of the same muscles and structures, but without the impact (peak loads) that running applies. And good walkers make good runners.
The pace you should strive for depends on your height and leg length; however, it should be punchy, but comfortable and sustainable. You should take long strides and keep the legs turning over rapidly. Your target speed should be between ~6.5 km/h (9.23 min/km) if you are ~150 centimetres (5') and ~8 km/h (7.5 min/km) if you are ~183 centimetres (6') tall. Different terrain (including uneven ground) and hills should be included between regular bouts in order to develop additional strength in the muscles and joints.
Only once you can comfortably maintain your target pace for five kilometres or more, should you begin to alternate bouts of walking with running. Start with shorter distances, and lengthen them gradually as you feel the body is able to keep up. Skipping (jump-rope) can be useful as a supplement to further develop joint strength—particularly in the ankles.
Needless to say, there are any number of ‘correct’ ways that you might approach this problem, but I hope this suggestion provides you with another perspective.