The first thing I'd do is verify - is she really slow, or is it just her perception that she's slow relative to others? You may already know the answer to this question... but I would start by talking to her teacher and seeing if her teacher feels that she writes slowly, isn't completing assignments, and if there is anything else going on at school - is she frustrated, happy, having other troubles etc.
Next step: you can check her writing speed at home. Have her write the alphabet (upper and lower case), and time her while she's writing it all out. Calculate her rate in letters/minute, and then google letters/minute + 3rd grade to find ranges that are typical for her grade. Also look at her writing - does she form all her letters correctly? Does she form them the same way every time? What does her posture look like while she's writing? Does she bend over or hold her wrist? Does she have a normal pencil grip?
If everything seems within range for her grade and she doesn't have issues with pencil grip or making letters etc... ask her what is difficult about writing things down.
Once you've done the above, if you still have concerns, I'd talk to her dr. You can have issues like this evaluated either through school or privately, and I've found that my children's dr was an invaluable resource in that he'd talked to enough parents of young children struggling in school that he had a good understanding of what we'd be able to achieve by going through an eval at our school vs a private eval. Whichever you choose, the important thing is to get the eval before you try to remediate the problem on your own.
Best wishes,
polarbear
