Roleplay it.
I.e. - drop most of the artifices of space combat as written and make up things as you go along. Just like any other aspect of RP.
Drop the turns thing - that really should just amount to players stating their intentions and a ref taking his 'turn' describing the results, counter actions and other goings on. Sure, fixed mount fighters may make multiple passes at targets, and energy weps might have recharge times, but that is the closest space combat is likely to come to 'turns'. Keeping track of time is a simulation concept, more than a roleplay one. It may sound unnatural, but normal roleplay of, say, a meeting with a patron - does that involve turns?
Most roleplayers are used to describing actions that occur in tandem and in parts. Of course, if a player decides that he is going to have his ship shoot out the bridge, all the turrets and whip around and aim for the M-Drive all at once, the Ref should put the brakes on. Which is real easy - require separate rolls dealing with them one at a time and allowing their opponents to do the same - with full knowledge of the players intentions. In all likelihood the player's plan will disintegrate and he'll learn its to his advantage not to 'telegraph' his moves to his opponents.
Drop 'Initiative' - just roleplay who fires first. Who gets the higher success (rolls) should determine any needed 'who got shot first' issues. Again, dealing with simultaneous action should not be difficult unless the number of combatants (or weapon elements) have gotten too numerous. Then you should resort to abstraction on a large scale - i.e. one roll may cover several ships, or a full action ala barrages, but probably in full rather than one at a time.
Speaking of rolls - as a game, its good to keep rolling dice. But the fiddly accounting that is part of most RAW combat mechanics, well, most folk rarely find anything good about that. If two fleets are fighting, its really kind of silly to roll for each weapon barrage for each ship. Group
actions and roll for degree of success.
Which brings me to the fact that you didn't mention what rule set you are starting from...?
(Now that was the 'quick' - for the cinematic, I track damage to the ship in more abstract terms and allow for aimed shots.)