Let's break the question into three parts, a map of 1800, 1810, and 1820. In 1800, which is about a quarter of a century after the American Revolution, there were sixteen states. The states were primarily converted colonial territories on the Atlantic Coast. The population of the United States according to the census was about 5.3 million. In 1800, France controlled most of the areas west of the Mississippi River, and Spain maintained a presence in what is now Florida as well as the southwest, in what is now Texas. The majority of America was still considered unexplored territory. Lewis and Clark would not be commissioned to explore the region in middle America until 1804.
By 1810, the census count rose to approximately 7.2 million. The United States added one state, bringing the total to seventeen states. The Louisiana Purchase in 1803 more than doubled the size of the territories in America ready for exploration and future statehood. The west, from modern-day Texas to what is now California, was still considered foreign land, but this was gradually changing as American pioneers slowly migrated west of the Mississippi River and began to establish settlements.
The census count of the United States by 1820 rose to approximately 9.6 million people. The United States had 23 states. The United States controlled all of the territories west to the Pacific. Looking at the map, you can see the parts of middle America, modern-day Texas, New Mexico, Arizona, Nevada, and parts of Colorado and Utah were still considered foreign territories.
The expansion of the United States west of the Mississippi River was due to the Louisiana Purchase and the gradual displacement of the Spanish in the southwest, bordering modern-day Mexico to the Pacific. Census counts provide evidence of expanding statehood and western expansion.