How does error increase when polynomial regression is applied?

Linear regression

library(ggplot2)
library(ggplot2)

set.seed(0)
x = 0:10
data <- data.frame(x, y = x^2 + 10*rnorm(length(x)))

ggplot(data, aes(x, y)) +
  geom_point() +
  geom_smooth(method="lm",
              formula="y ~ x - 1") +
  ggtitle("y ~ x - 1\n")

When I have only one free parameter, x (and no intercept!), then standard errors increase linearly with the size of x.

Polynomial regression

ggplot(data, aes(x, y)) +
  geom_point() +
  geom_smooth(method="lm",
              formula="y ~ I(x^2) - 1",
              color='black') +
  geom_segment(aes(x=5, xend=5, y=23, yend=27), color='red') + 
  geom_text(x=5.2, y=25, label="~4", color='red') + 
  geom_segment(aes(x=10, xend=10, y=92, yend=108), color='red') +
  geom_text(x=10.25, y=100, label="~16", color='red') +
  ggtitle("y ~ I(x^2) - 1\n")

When I have a polynomial free parameter, x^2 (still no intercept), error rises quadratically with x. It is 4 at x=5 at x=10. X has gone up 2 fold, error has increased 2^2=4 fold.