It's the way this game works.

1) Casting time, if applicable.
1a) At cast start (first 0.5s), checks for facing. If improper, cancels.
1b) At cast end (last 0.5s), checks for facing. If improper, cancels.

2) Cast finished (or in the case of instant-uses, checked for facing and said "ok")
-At this time, Damage, effect or no effect, invulnerability, whether a target is in range for AoE, etc are all checked. If the user of the skill dies before the following steps complete, it is all cancelled (notable in cases of long animation time like Melee LB2 or Caster LB2/3)

3) Animation time.

4) Damage and effect pop-ups ("Fire 510", "+Bio", "Bind NO EFFECT", etc). If the attack inflicts knockback, knockback direction is determined at this step.

5) Effects applied, damage dealt is changed on the HP bar

There is a subtle but distinct delay between all of these "steps". It's just how the game works.

Skills like Blunt Arrow, Savage Blade, and Overpower have a near-instant animation time so the delay is not as noticeable. Skills like Holy, Rage of Halone, and Steel Cyclone have significant animation delay.