The biggest growth actually happened in little more than 100 years during the 19th century and early 20th century and this was largely the result from a massive emigration of different Europeans which were victims of constants famines which struck the 19th century, alongside religious/political prosecutions, national revolutions which plagued Europe after the French Revolution, and wars. Europeans left the continent in very big numbers just like Syrians are doing now from their homeland, basically.

And, where to head? Well, the Americas was by far the most attractive prospect because it was barely populated at all and because of that full of untapped natural resources which both guaranteed abundance in food and work opportunities. Most the governments in the Americas were far more liberal and less strict in its policies of control since they were recently funded states with constitutions without imperial institutions, which also promoted immigration-friendly policies just like gifting hectares of farmland, tools and other kinds of assistance. They promoted immigration because they were barely populated states and as a way to increase their power and influence they had to grow their numbers, too.

Another factor has to do with the topic of slavery. As businessman wanted to tap on the untapped resources they needed labor, a lot of it. The massive influx of immigrants peaked in the second half of the 19th century, but for the first half and earlier times, many nations and states satisfied the need for labor by importing hundred of thousands of slaves from Africa.

And I mention "the Americas" and not just the United States since this was a phenomenon that struck the whole continent! Latin America, specially the far southern states like Argentina or Brazil also received a massive influx of immigrants which multiplied many times the population in a mere century!

After WWI the massive influx stopped as Europe began to adopt ammoniac fertilizers which greatly limited the scope and number of famines, religious prosecution relaxed, many empires just outright ceased to exist and with it an expansion of personal liberties, further economic expansion and so on. And since that point most the Americas have just kept their pace with the growing world population. The United States whites began to develop lower fertility rates though, but this has been compensated by a never-ending influx of immigrants pursuing "the American Dreeam" that continues even today, does compensate it.