I have a thick white line down the length of my big toe that  grows as the toenail grows?

Sometimes. fungus takes the shape like that but it doesn't have to be. A foot doc or dermatologist sometimes can biopsy the toenail if they feel it is warranted.
Fungal infection? If you did not hit your foot it sounds like you may have a fungal infection. Usually when it is white it is usually due to the moisture loving fungus Candida... Good news it is pretty easty to get rid of so call your podiatrist.
Injury or fungus. You may have injured the toenail or you could have a fungus.
Toenail line. You probably had some trauma in the past to the growth plate which nows constructs the toenail with the line. This might be unsightly, but unless thick and uncomfortable, can be ignored. It might be an excuse to get a pedicure with polish if that fits your lifestyle.