Turn based movement is an ease of play abstraction. It has enormous numbers of difficulties, which a DM ought to occasionally be willing to deal with by on the fly rulings whenever the rules are breaking player emmersion and creating low fidelity.
However, in general the problems created by turn based resolutions are smaller, much much smaller, than the problems created by attempting to create high fidelity less abstract resolutions.
It's pretty easy to increase the accuracy of a turn based resolution system with respect to capturing analog movement. You simply increase the sampling rate by decreasing the amount of time covered by the abstract round. Often this ends up involving a multi-tiered timing system with larger turns and smaller rounds, segments or impulses. The smaller the amount of time covered in a round, the smaller the 'squares' you can occupy, then the greater fidelity you will have to realities continuous analog experience. However, this tends to be chasing something down the rabbit hole. The more you try to make the actions in a round concrete representations of reality, the more you tend to slip into an 'uncanny valley' where the lack of adherence to expectation more and more annoys. Worse yet, the time required to resolve the combat increases exponentially. Instead of resolving combats in 2 or 3 or 5 or 20 rounds, you may well need 100 or 200 rounds for a combat involving very careful time tracking as actions will typically be ongoing over a different number of rounds for each actor. You might start a 5 round action in turn 6, while the other player has started a 3 round action in turn 4, the spellcaster has a 6 round action that they started in round 2, and the monster is in the midst of a 10 round action begun on round 1. This involves a lot of mental overhead and bookkeeping, and tends to make combat drrrraaaaaggggggg and bog down your story as combats take up a greater and greater percentage of play time. Ironically, what may have started with the intention of having more exciting and more interesting combats, will often result in rules sets that encourage players to avoid combat as a resolution method simply because of the crushing time burden it imposes.