The turnip being the most popular root vegetable is certainly true; turnips were widely grown in nearly all parts of Europe, not least because they're pretty tough vegetables once they get going, and nearly immune to many pests, and can be eaten by both humans and animals. And they keep pretty well, too.

However, it's probably not true that land used for potatoes in 1800 would have been under turnips four centuries earlier. There are two main reasons for this: grain crops were more prevalent than turnips ever were, and potatoes will grow in places nothing else will.

Pre-Columbian-exchange Europe ran on grain. Wheat, rye, oats and barley were grown, depending on where in Europe you were and to some degree what your social status was like. Wheat, used almost solely for bread, had the highest social status, but was also the most difficult from which to get a good yield. A return of 1:3 was pretty normal there. Rye and oats grow better in Northern Europe, so they were more common there. Compared to the amount of grain grown, turnips were a minor crop, and other vegetables lagged further behind. Grain crops require pretty decent land, though, so there are lots of places they won't grow.

Potatoes, though, will grow on marginal bog land in Ireland, halfway up the side of a French mountain, in forest clearings in Scandinavia, and so on. This meant that they could be added to the range of agricultural options without taking much away (in most places) from the grain harvest, and people could still grow other vegetables, such as turnips.

Sources:

C. M. Woolgar, D. Serjeantson, and T. Waldron, Food in Medieval England: Diet and Nutrition
Alfred W. Crosby, The Columbian Exchange: Biological and Cultural Consequences of 1492