The core accretion model is probably the most accepted model of planetary formation.

According to this theory, the planets formed near their current orbits from material found there. Material near the center of the disk would be too hot for light elements to condense, and solar wind would also sweep the lighter elements out to greater distances, so the outer planets should be lower in density than the inner planets. As indicated in the question, this is not the case in our solar system; Neptune and Uranus are more dense than Saturn. One of the more-accepted explanations is that of Neptune and Uranus formed in roughly the same region as Jupiter and Saturn, but then migrated out as a result of gravitational interactions.

Planetary migration is counter to the accretion model, but the discovery of planets which could not have formed in their current locations, such as Hot Jupiters, indicates that movement of gas giants is not implausible and may even be common. In addition, their are many theories for the formation of planets, with none being universally accepted.

The answer is that density is not the way that the planets arranged themselves.

Exactly how the planets did arrange themselves is still an area of active research. We now know of many, many Jupiter-sized and larger gas giants located much closer to their suns than even the orbit of Mercury, let alone Earth. Clearly, the configuration of our solar system with rocky planets inward, gas giants in the middle, and icy Kuiper objects like Pluto and Eris outward is not the only way that nature can do it. We still don't know just how (un-)unique our own solar system is, though.