Hi Katie, on top of what the others have suggested pay attention to how you're positioning your hand on the neck.
Imagine that you're holding a glass of water - look at the shape your hand takes on, you'll notice that your fingers are curled but comfortable. Using this shape position your thumb on the centre of the back of your bass's neck, even with smaller hands this should give you adequate reach to play comfortably.
Don't worry if you can't cover 4 frets (ie 1 finger per fret) - this will come with time and practice.
Here are a couple of links you may find helpful.
The first is an excellent article by Matthew Brown here on AB
Position is Everything
The other is to Adam Nitti's website - review all of the lessons but in particular the one on left hand position..
Adam Nitti Lessons
Most importantly - be patient with yourself & have fun!