Do a basic attack with your sword. See that big animation of you swinging your sword? That takes time, but basic attacks themselves are instant. The underlying combat mechanic is an instant damage attack, but the animation of the attack is not instant, obviously. Do your basic attack, then immediately follow it up with a magicka/stamina-based skill. Since the basic attack was instant, it's combat mechanics are finished and the game can prioritize your skill attack over the animation of the basic attack. So you can do more damage in a shorter time.

Basically, a lot of the animations in the game are purely cosmetic. You can bypass them if you initialize another action after the damage is done but before the graphical fluff finishes.

Practice your timing for starting a basic attack, letting go, triggering the skill attack, and working that into a rotation. Get a feel for what takes priority and what will cancel an animation. It takes some work to get it right, but you will get it set to muscle memory eventually. You will know when is too soon, when you have waited too long, etc.