They are probably reaching for more light. In your area in the winter with shorter days that could happen. In summer they will burn with the hot sun and get white or tan spots.
Does it look anything like the link below of the my little newbies a couple years ago. If so you should be ok.

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I think you are right--the weak light is the problem. Is it okay to trim off the longer leaves? What is the best way to encourage new leaf growth?

How long are the leaves? Can you post a pic?
The best way would be to provide the ideal conditions for growth. I know that isn't the answer you want to hear.
They like bright light, somewhat moist, not wet.
You could feed it a mild fertilizer if you think it has good roots.
Most plants that seem elongated are reaching for light.

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Yes, unless you are able to provide your plant with an improved light situation, you'll never be pleased with the results. Healthy, vital plant growth requires that ALL of the pieces of the puzzle are in place, not just some of them.

Thanks for your input. Is placing it directly in front of a south facing window not giving it enough light? The reason I ask is that the new growth has been much healthier. The long spindly stems were already on the plant when I received it. I got it at the end of last summer, and then of course it didn't grow much during the winter. Since this is the first really good growing season that I'll have the plant, I just wanted to give it the best chance possible to bush out a little more.

Unfortunately I don't have any pics right now. The elongated leaves were on approximately 8 inch stems with tiny leaf clusters at the end. It's grown a few new stalks since I've had it, which are more in proportion--stems are about 3-4 inches long, with proportionate leaf clusters. I've pruned a good bit of the long ones off, but it hasn't put out new growth were they were. Should I go ahead and do a drastic prune? Will it fill in the pruned places with new leaves?

You will want to be careful later in summer in front of the south facing window. For now it might be ok, but probably not past May depending on where you are. Then you'll want to put a few few away from window just off to the side, so no direst sun.
Pruning off the old ones to my knowledge won't put out new growth in the old spots. If you are going to do a drastic prune now would be the time. Wherever you cut the main stalk it will put out new growth just below it. You can strip off all the leaves and cut into pieces and root like I did above in moist soil. It might take awhile. Rooting hormone helps.

Thanks for the feedback. I think the plant must be the pony tail palm as others suggest. I really hate to lose the plant, so maybe I'll put off doing anything for now, unless others have had better luck! Thanks.