The speed of any object is a balance between the drag force on the object and the thrust the object can create. Attaining high speeds, or possibly more relevant in this case, high acceleration requires making the thrust as high as possible while keeping the drag as low as possible.
Water is a lot denser than air, but this affects the drag mostly when the flow is turbulent and the drag is dominated by inertial forces. In this regime the drag is effectively due to having to push the medium out of the way, and it's easier to push aside low density air than high density water. However if you can keep the flow laminar the density isn't as big a factor and the drag is dominated by the viscosity of the medium. Water is a lot more viscous than air (as well as denser) but for streamlined objects the drag due to viscosity can be kept remarkably low.
Where water wins is that it's much easier to develop a high thrust in water than in air. In a fluid medium, where there is nothing solid to push against, you produce thrust in basically the same way that a rocket does. If you push away some mass of water $m$ with a velocity $v$ then the momentum of the water changes by $mv$, which means that your momentum changes by $-mv$. So you push the water in one direction and you accelerate in the other direction. The thrust you generate is simply the rate of change of momentum of the water.
And it should now be clear why it's easier to generate a high thrust in water than in air. Because air is low density you can't push a high mass of it (unless you're very large) so it's hard to change its momentum by very much.
So to summarise:
How the speeds in water and air compare depend on the exact tradeoff between drag and thrust. Sailfish can reach speeds of 68 mph, but they do this mainly by being very streamlined so they can keep the drag as low as possible while exploiting the high thrust they can get from water. Birds generally don't reach speeds this high because although the drag in air is small they simply can't generate the thrust required for high speeds. Peregrine falcons can reach speeds of 200 mph, far faster than a sailfish, but they do it only in dives where gravity provides the thrust.