Rewatch the scenes from the Battle of Wolfe 359 in the premier episode of DS9, they have an Ambassador-class and a Nebula-class ship (IIRC) making hard 90 degree turns to attack the Borg cube. It’s not the first time you see a StarFleet ship doing some impressive maneuvers.
In fact, watch the attack on the Borg ship in First Contact as well. IIRC, there were some fancy maneuvers in there too.
Ruger
DS9 version of wolf 359:
https://www.youtube.com/watch?v=sgAlog2eTEgand the enterprise does some pretty zippy maneuvers at times in TNG, usually just in brief scenes where they turn around to leave a meeting with another ship.
a good, though less noticable example is in "booby trap" where picard flies the Ent-D agilely through an asteroid field using only one thruster pack at a time.
https://www.youtube.com/watch?v=hycMgv8nSO0given the sheer mass of the galaxy class, those have to be extremely powerful thrusters.
the main reason the Ent-D rarely seemed to maneuver much in fights was just the nature of the battles.. they were usually either things that started during negotiations where both side's ships were stationary when the shooting began, or were ambushes where picard/riker was trying for non-violent resolutions, and only resorted to weaponry as a last resort. (or in the case of yesterday's enterprise, was tied to a specific location and acting to shield another ship from harm, preventing complex maneuvering.
https://www.youtube.com/watch?v=igF62y6dlJk )
of course all of this was the development of the modelwork camera rigs and compositing technology over the course of TNG allowing for more impressive VFX.. and DS9 coming in at the end of that process for model work, and then pioneering the use of computer graphics in trek shows as an alternative to physical models. (that DS9 sopent the first several seasons with much smaller and more agile
looking craft, the runabouts, probably helped.. no one expects a massive ship liek the galaxy class to pull hairpin turns, though it did sometimes. but a runabout is small and feels more like a fighter than a battleship. then they applied that style of the defiant.. a battleship that flies like a fighter.

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