Recently I tried to fit some points which (from the plot) seems linearly distributed. 
The fit result (in R) is:

    Residuals:
        Min      1Q  Median      3Q     Max 
    -112223   -2532    2021    3698   83241 

    Coefficients:
                  Estimate Std. Error t value Pr(>|t|)    
    (Intercept) -6.623e+03  7.136e+02  -9.282   <2e-16 ***
    population   5.946e-02  4.278e-04 138.986   <2e-16 ***
    ---
    Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

    Residual standard error: 12780 on 379 degrees of freedom
    Multiple R-squared:  0.9808,	Adjusted R-squared:  0.9807 
    F-statistic: 1.932e+04 on 1 and 379 DF,  p-value: < 2.2e-16

With $R^2$ 0.98. Nice! BUT I checked the OLS assumptions and from the graph I have heteroscedasticity (top-left and bottom-left graph) and I have no normality of errors (top-right graph).

[![enter image description here][1]][1]

So I transformed the dependent and independent variables with log transformation. I now have a model which meet all the assumptions but it is more complicated (exponential fit?) and with lower $R^2$: 0.96.

    Residuals:
         Min       1Q   Median       3Q      Max 
    -0.37058 -0.06061 -0.00701  0.05532  0.44428 
    
    Coefficients:
                   Estimate Std. Error t value Pr(>|t|)    
    (Intercept)    -2.08467    0.06153  -33.88   <2e-16 ***
    log_population  1.12146    0.01120  100.12   <2e-16 ***
    ---
    Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
    
    Residual standard error: 0.101 on 379 degrees of freedom
    Multiple R-squared:  0.9636,	Adjusted R-squared:  0.9635 
    F-statistic: 1.002e+04 on 1 and 379 DF,  p-value: < 2.2e-16

Which is the best? Is the first model wrong, and why?


  [1]: https://i.sstatic.net/BBh0e.png