I'm playing BGI as a fighter but I have a few mages (dynaheir and viconia) and they only have around 4 or 5 spells they can cast.

Once I cast the spell, I can't cast it again until I rest?
How am I ever supposed to cast spells quickly and multiple times?

Is there something I'm not doing correctly?

That's the way it works in 2nd edition D&D - low level mages are weak as kittens.

Give them slings and let them sit at the back plinking away.

Be judicious with your spell casting - use spells only when really necessary.

The sorcerer class in BG2 (and presumably TUTU, et al) added a bit more spell casting ability at the expense of a smaller spell list.

then how does anyone play a solo mage game in baldurs gate?

By knowing the game backwards, forwards and sideways. Every location, character, quest, trap and monster.

Since a single character gets all the XP, the player can pick the easy quests early on and build up levels. Enough levels and the bread-and-butter spells (eg fireball) can be cast - then the mage starts to shine....

See here: http://www.gog.com/en/forum/planescape_torment/what_am_i_missing_magic_characters_seem_helpless_in_dd_games

similar question.

Short answer: you're not supposed to cast spells quick, at least at the beginning. You're supposed to keep the mages as backup for when more firepower is needed. Or crowd control. You get CC spells early on and significant damage only later.

The mage and spellcasters are definitely the most fun things about D&D. They might suck at low levels but wait till they reach high levels with a wide array of spells to choose from, that's the point where they literally kickass.

Mages are not limited to spellcasting, they are allowed to fight physically too.

To reiterate what's already been said: At low levels skip direct damage spells and just disable. Allow higher level mages to do the nuking.

You can use ranged weapons like darts, sling and throwing daggers when you run out of spells.

Sleep is one of the most powerful spells for a low level mage and can easily take out 4-5 low level monsters.

Study the same spell multiple times. Such as Magic Missile three times