Let the upper-left corner be $(0, 0)$ and the bottom-right corner be $(2020, 2020)$. Claim that the answer is $1515$, and the construction is initially painting $(505, 1010), (1515, 1010)$. Suppose the opposition that the answer is less than $1515$. That is, for any square, the minimum Manhattan distance from the square to an initially painting square is at most $1414$, and we will say that a square is controlled by a square iff their distance is at most $1414$. Consider the $8$ squares: $(0, 0), (1010, 0), (2020, 0), (2020, 1010), (2020, 2020), (1010, 2020), (0, 2020), (0, 1010)$, there are no $4$ squares that can be controlled by the same initially painting square since in every $4$ squares there are $2$ squares with Manhattan distance $\geq3030>1414\times2$. $\therefore$ it is impossible to finish in $1414$ steps.