How to Understand Turbo Lag
Turbos rely on the flow of the exhaust gas through the turbine housing. As the engine RPMs increase the turbo spools and begins to build boost. The delay between the opening of the throttle and the turbo spinning and producing boost is known as "turbo lag".
Typically, a turbo with a physically small turbine and compressor wheel will have low inertia and therefore accelerate faster for a given exhaust flow. However, because of their small size, the amount of boost they can produce will be limited at higher engine speeds. Conversely, a turbo with a physically larger turbine and compressor wheel will have higher inertia and will take longer to "spool up" or get to speed. With increased lag however the trade-off is increased boost at higher RPM.
On average a small turbo can deliver good power early on but run out of steam pretty quickly. Similar to a supercharger. While a larger turbo will give a fair kick when the boost comes on and will hold that through the rest of the rev range. This can be an issue because it can make a car difficult to drive.
Various methods of minimizing turbo lag have been attempted over the years. To get a combination of boost pressure and hence power along with fast throttle response and drivability. Mazda used a sequential turbo on the last of the RX-7 models, with a small turbo spinning at low to mid-range engine speeds and then a bigger version taking over at high speeds.
The current crop of World Rally cars probably have the most efficient system, known as anit-lag (ALS). Details vary from manufacturer to manufacturer but they all work on the same basic principle. A small amount of compressed charge air can bypass the throttle when it's closed. The ignition spark in retarded at the same time and un-burnt fuel is dumped directly into the exhaust turbine housing, where it mixes with the air and is spontaneously ignited by the red hot wheel. This keeps it spinning so that as soon as the throttle is opened, the turbo's ready to give full boost. Because the throttle is closed during the above process the inlet tract is filled with compressed clean air, so that when the throttle is opened, the cylinders are immediately filled and when the fuel is injected, response is instant. The downside is the ALS process is the massive thermal loads it puts on the turbo and exhaust system, and the heat generated in the engine.
Another method to reduce lag (partly) when changing gear is a dump valve. Normally on a gearshift the throttle is closed and the air from the spinning turbo backs up against the closed throttle and stalls it. But a dump valve vents the pressurized air off.