So, this question lies kind of the periphery of things that I know about—meaning that I can offer you an answer that is based in scholarly work and historical evidence, but I have just a little bit of anxiety that someone who really knows their American agricultural history is going to jump off the top rope with a lethal "well actually." Maybe I'm being too cautious with that caveat, but I thought I would offer it up anyway and let the mods/gods decide.

First off, I will say that soil erosion and soil exhaustion have been issues in many different places at many different points in time. I know next to nothing about Asian agricultural history, but I would be extremely surprised if there were not periods and regions in which farmers overworked their land, and that this resulted in mass migrations.

I also think that it is important to note that calling land "exhausted" or "ruined" is kind of a relative term, one that has as much to do with economic conditions as the nutrient content of the soil. For example, if crop prices are high, operating costs are low, and the mortgage on your farm has favorable interest rates, you might be able to farm relatively infertile land quite profitably. If crop prices drop, however, or the rain doesnt come, or the banks go under, even fertile lands might become financially unfeasible to farm.

That said, it is undoubtedly true that erosion and depletion of soil nutrients were major issues in the 19th century United States. We know this, because a great many Americans, and foreign travelers in the United States, pointed this fact out. Starting in the late 18th century, many different writers criticized American farmers for overworking the soil. William Strickland, an English farmer who visited the United States in 1795 would write, “the land owners in this state are, with few exceptions, in low circumstances; the inferior rank of them wretched in the extreme… Land in America affords little pleasure or profit and appears in a progress of continually affording less…Virginia is in rapid decline.” George Washington, an acquaintance of Strickland's, agreed, writing in a letter to him that “the system of agriculture (if the epithet of system can be applied to it), which is in use in this part of the United States, is as unproductive to the practitioners as it is ruinous to the land-holders. Yet it is pertinaciously adhered to. Our lands… were originally very good; but use, and abuse, have made them quite otherwise.” The problem was especially bad in the south, where much of the rural economy was based around growing monocultures of tobacco, and later cotton, for export. Tobacco depleted the soil of nutrients quickly, and farmers resisted applying manure to their fields, believing it ruined the flavor of the leaf. Instead, many grew tobacco for only three or four years, at which point it was no longer economically feasible and they then switched to was what is typically called the “three-crop system.” Under this method, farmers would cycle through planting corn one year, then wheat, then resting their field for a year. This rotation, although initially productive, eventually exhausted the soil, depleting its nutrients and resulting in smaller and smaller yields. Finally the land would give out entirely, at which point they would abandon it for twenty years or more. One contemporary of Washington's called this pattern an “exhausting rotation of crops” and a “land-killing system.” Even the resting of fields, intended to restore fertility seemed counterproductive. Without cover crops the topsoil eroded and fields suffered from the accumulation of “binding Weeds and Rubbish–and the Hoof that beats it to a dead Closeness,” in the words of one Maryland planter.

So what explains this? At the time, some attributed this overuse of the land to ignorance on the part of American farmers, or to some defect in the national character. This point of view was particularly common among English observers, who were inclined to view Americans as upstarts, hillbillies, and unrefined frontiersmen. But many also recognized that there was something more fundamental at work. The economic conditions in the United States were very different from those in Europe. Compared with Europe, the United States had a seemingly endless supply of uncultivated land, as long as Native Americans could be dispossessed of it. This open frontier acted as a kind of population sink, drawing potential agricultural labor away from more established areas. The situation in England and much of Europe was the exact opposite: Land was scarce and expensive, while labor was plentiful and cheap. Consequently, European farmers could better afford to apply labor-intensive farming methods that preserved soil fertility, while many Americans had to quickly extract as much value as they could from a plot of land, before abandoning it and moving on. Washington described this dynamic in a letter to the English agriculturalist Arthur Young, “The aim of farmers in this country (if they be called farmers), is not to make the most they can from the land, which is, or has been cheap, but the most of the labor, which is dear; the consequence of which has been, much ground has been scratched over and not cultivated or improved as it ought to have been.”

Not even wealthy, slave-owning planters like Washington were spared from this pattern. For those who already owned land, the availability of unsettled territories depressed its value, while simultaneously inflating the value of the labor they needed to farm it. This drove landowners, or their tenants, to farm the land ever more intensively, depleting its resources and further diminishing its value. Paradoxically, the practice of slavery contributed to the problem as well. One might expect that slavery, which provided planters with a labor force incapable of immigrating westward, would be a potential solution to this pattern. In practice, however, just the opposite was true. Since the cost of owning slaves was fixed and unrelated to the labor they performed, slaveowners were incentivized to keep them constantly at work. As a result, southern planters were constantly expanding their fields, or planting crops in land that had already been exhausted. Like northern farmers, southern planters were obsessed with maximizing the productivity of their labor force, rather than the productivity of their land. If the land in Virginia or the Carolinas gave out, they could always relocate their slaves to further west and repeat the process over again.

In this sense, there is kind of a "chicken and the egg" quality to OP's question. It is true that the exhaustion of eastern soils at times helped to drive westward expansion. At the same time, the possibility of western expansion drove up labor costs and drove down land costs, thereby incentivizing American farmers to use the land ever more intensively.

Most fascinating and educational thread I've read in.... forever. Thank you!

Plus, a new book recommendation!

I don't agree with the premise. Farming spreads were relatively cramped in New England and New York State and Pennsylvania because of the terrain. The northern and central Middle West offered more arable land for a good price, so farmers and homesteaders sold their plots in the East and moved to Iowa, Minnesota, Wisconsin, Michigan, etc. Their old farmlands were not exhausted or eroded, and many are being tilled to this day, though they are much smaller holdings than the western farms.

You may be thinking of cotton planters in the South and Southwest, c. 1820-1860. In the days before fertilizers and mechanical tilling, cotton was a crop that could use up the soil in a few seasons. And so planters did move west and develop new holdings, as far west as Texas. But that's an exceptional example and does not reflect farming in general.