I am currently really struggling with my pure Mage and I am using Lydia as my companion. I have her using a axe and shield. I am only lvl 8. I am using Lightening as my main, but fire against the undead. I am currently struggling against packs of draugr, If there is more then one wight and/or scourge I am SOL. I have tried everything from summoning to using lesser turn undead. I can't get turn undead yet, I am too low. Lesser Turn undead only does up to lvl 6 and I think my opponents are above that, so it is no help.

Assuming your in vanila game, Lydia is not a good choice if your character is trying to sneak up. By default , she wears heavy armor & its too noisy for this. You will also find destrucition magic is difficult to level up in this game without training and associated perks. Have your tried conjuring up entities like flame attranoch or something? this can work sometimes ok, but don't rely on it too much. Better to spread skills over several categories imo.

At lower levels you need to be more careful when playing a pure mage. Don't charge in and attack multiple enemies and expect to survive. Get a spell to heal you companion so they can take more attention from the enemy.