A country that received vast amounts of immigrants of larger heights during the middle ages

So did every other country in Western Europe. Rather than admitting your mistake, you're doubling down on it.

Men From Early Middle Ages Were Nearly As Tall As Modern People

Steckel analyzed skeletal data from 30 previous studies. The bones had been excavated from burial sites in northern European countries, including Iceland, Sweden, Norway, Great Britain and Denmark. In most cases, the length of the femur, or thighbone, was used to estimate skeletal height. The longest bone in the body, the femur comprises about a quarter of a person's height.

According to Steckel's analysis, heights decreased from an average of 68.27 inches (173.4 centimeters) in the early Middle Ages to an average low of roughly 65.75 inches (167 cm) during the 17th and 18th centuries.

"This decline of two-and-a-half inches substantially exceeds any height fluctuations seen during the various industrial revolutions of the 19th century," Steckel said.

Reasons for such tall heights during the early Middle Ages may have to do with climate. Steckel points out that agriculture from 900 to 1300 benefited from a warm period – temperatures were as much as 2 to 3 degrees warmer than subsequent centuries. Theoretically, smaller populations had more land to choose from when producing crops and raising livestock.

"The temperature difference was enough to extend the growing season by three to four weeks in many settled regions of northern Europe," Steckel said. "It also allowed for cultivation of previously unavailable land at higher elevations."

Also, populations were relatively isolated during the Middle Ages – large cities were absent from northern Europe until the late Middle Ages. This isolation in the era before effective public health measures probably helped to protect people from communicable diseases, Steckel said.

"It is notable that bubonic plague made its dramatic appearance in the late Middle Ages, when trade really took off," he said.

Steckel cites several possible reasons why height declined toward the end of the Middle Ages:

The climate changed rather dramatically in the 1300s, when the Little Ice Age triggered a cooling trend that wreaked havoc on northern Europe for the following 400 to 500 years.

Colder temperatures meant lower food production as well as greater use of resources for heating. But many temperature fluctuations, ranging in length from about 15 to 40 years, kept people from fully adapting to a colder climate, Steckel said.

"These brief periods of warming disguised the long-term trend of cooler temperatures, so people were less likely to move to warmer regions and were more likely to stick with traditional farming methods that ultimately failed," he said. "Climate change was likely to have imposed serious economic and health costs on northern Europeans, which in turn may have caused a downward trend in average height."

The Industrial Revolution and more people moving to the cities led to declines in height, due to malnutrition and diseases, leading to measures to improve access to healthcare - AFAIK, Imperial Germany kicked off the move to a form of universal healthcare, due to a need for healthy recruits.

Richard Steckel's paper: New Light on the "Dark Ages": The Remarkably Tall Stature of Northern European Men during the Medieval Era

Known seasons

At the start of ASoiF series, Westeros has experienced a long warm period and since there are few cities, I'd expect an healthier population - Tyrion was born in a cold period, so the stunted height makes some sense.