Essentially, this is a result of observational bias. A natural satellite will only orbit a parent for extended time periods precisely because the orbit it is in is stable †.
The plain truth of the matter is that we are simply injecting satellites into unstable orbits. If you were to move natural satellites into the same orbits, they'd be unstable too.
Take for example, the moon:
Gravitational anomalies slightly distorting the orbits of some Lunar Orbiters led to the discovery of mass concentrations (dubbed mascons), beneath the lunar surface caused by large impacting bodies at some remote time in the past. These anomalies are significant enough to cause a lunar orbit to change significantly over the course of several days.
From Lunar Orbit.
What this is saying is that there's no (or more correctly, very few) stable orbits around the moon due to its lumpy gravitational field. Why do we not see natural satellites orbiting the moon? Because they would have decayed due to the orbit being unstable!
Another example, asteroid 3753 Cruithne.
3753 Cruithne is an Aten asteroid in orbit around the Sun in 1:1 orbital resonance with Earth, making it a co-orbital object. It is a minor planet in solar orbit that, relative to Earth, orbits in a bean-shaped orbit that ultimately effectively describes a horseshoe, and which can transition into a quasi-satellite orbit.
Its orbit, is too, unstable. On the timescale of millions of years, it will transition out of its current arrangement too:
After many years, the Earth will have fallen so far behind that Cruithne will then actually be "catching up" on the Earth from "behind". When it eventually does catch up, Cruithne will make a series of annual close approaches to the Earth and gravitationally exchange orbital energy with Earth; this will alter Cruithne's orbit by a little over half a million kilometres—while Earth's orbit is altered by about 1.3 centimetres (0.51 in)—so that its period of revolution around the Sun will then become slightly more than a year.
But there's even natural examples of unstable orbits on human timescales. Look at 2006 RH120
2006 RH120 is a tiny near-Earth asteroid with a diameter of about 2–3 meters that ordinarily orbits the Sun but makes close approaches to the Earth–Moon system every twenty years or so, when it can temporarily enter Earth orbit through temporary satellite capture (TSC). Most recently it was in Earth orbit from September 2006 to June 2007.
But, we can also inject artifical satellites into stable orbits, as well. Now that Dawn is orbiting Ceres, it will stay there for many hundreds of years. You could easily consider that stable on human timescales.
To summarize, orbits don't care whether the bodies involved are artificial or natural. You're only likely to find natural satellites in stable orbits, because the chaotic nature of orbits takes place over geologic timescales.
†From a physical perspective, no orbit is ever stable. Tidal effects and gravitational influences mean that most of the orbits we consider stable, on human timescale, are unstable in geologic timescales. Additionally, gravitational radiation results in orbital decay on a timespan longer than the scale of the universe. All orbits are merely a human approximation of an unstable, chaotic system.