Trier Germany has it right, and anyone claiming that turbos by definition build power to redline have it REALLY backwards and need to look at some dyno charts to get sorted.
Turbos sized by OEMs to minimize lag (e.g., nearly every current turbo car, including BMW's N54 / N55, VAG 2.0T, etc.) are small... meaning they spool up quickly (low inertial mass) but consequently cannot produce high boost at high rpm. Racing turbos are usually the opposite, huge in size meaning they are "laggy" but can produce huge HP at high rpm / redline.
But putting aside mechanical theory, how can anyone who has driven an N54 / N55, the MINI R56 turbocharged Prince 1.6, or the VAG 2.0T family of engines not notice that the power delivery drops off around 5500rpm? It is terribly obvious, and a turnoff to many enthusiasts. The low-end and mid-range grunt is fearsome, no doubt about that (go test drive the Audi 2.0T in the B8 A4 to see what massive low-end and midrange torque is all about!), but the "rush to redline" just isn't there with most OEM turbo applications. Hence why many are NA engine enthusiasts... it isn't about being "pure", it is a characteristic power delivery that many enthusiasts find far more rewarding than having grunt off idle but then having to shift up before 6000rpm.
If anything, the current crop of OEM turbo motors feel a lot like old school OHV V8's by the American manufacturers... if you like that sort of thing, you're in luck! But know that the American V8's have moved on from that type of power delivery and now have a "cammy" feel, high redline, and power that builds to redline. e.g., Ford Mustang.