Expedite
...and from a tower point of view.
If we are faced with a potentially tight gap to get a departure away, we will prep up the traffic as follows:
The first inbound could be asked "..after landing expedite vacating."
The departure will be told to be ready immediate and the second inbound may be warned to expect a late landing clearance.
What we mean by expedite vacating is minimum time on the runway. This should be the norm and many of you try your best to minimise your time on the runway, but that one occasion when you need somebody off quickly...!!! You get the picture. But in line with the previous posts, it should not be over used otherwise the word will loose its impact.
Different scenario, on the ground, as an example, you may be tight on your slot and asked to expedite your taxi. In plain enlish that translates to as quick as you can commensurate with safety.
In a nutshell I guess expedite to the tower peeps means 'without delay'.