Mine is an BMW M45 direct injection turbo diesel. In the early days of small turbos, such as the early SAAB, they suffered turbo lag - until the revs built up, the turbo wasn't very effective, so you got nothing, then a sudden kick in the back, as the turbo began to function.
More modern diesel turbo's, are much more gradual. Mine, in factory ECU trim, had a slight delay - quite intimidating, when trying to pull out into traffic and on roundabouts, was it going to pull or not. Reprogramming the ECU's MAP, somehow sorted that out, and the acceleration is now very predicatable, and progressive, plus a lot more BHP available to use.