As we can see $e^x=x^2$ has 1 solution.
$e^x=x^4$ has 2 solutions with $x^4 > e^x$ after $x=1.43$ which is one of the solutions.
What is the reason behind this?
Also what would be the approach is we were asked to find the number of solutions of $e^x=x^3$? If we see graph of $e^x = x^3$ we think it has 1 solution but it actually has 2 solutions $x=1.857$ and $x=4.536$, so why is there such variation in slopes?
As $x^4$ dominates over $e^x$ after some $x$ till infinity, while in $x^2$ and $x^3$, $e^x$ is dominating.
Non-constant polynomials will only dominate $y^x$ when $y \leq 1$ and since $e>1$ we know that $e^x$ will dominate $x^n$ eventually. Note that your axis only goes as high as $x=6$ which is extremely far from $\infty$ so you can't tell anything about the dominating behavior. In fact, no graph of finite size will ever give you any information about the limit at $\infty$ and you'll need to rely on calculations to evaluate it.
If you graph $e^n-x^n$ it's zeroes will show you when $e^x$ begins to dominate $x^n$ as it crosses the $x$-axis. Here you can just check that $e^{10} > 10^4$ directly or you can just zoom out on the graph so that the values at $x=10$ visible.
If you're familiar with calculus then you can see taking derivatives of $e^x$ will always give you $e^x$ but I can differentiate a degree $n$ polynomial $n$ times and will be constant. So this means the growth of $e^x$ will always eventually be faster than a polynomial and thus will eventually overtake $x^n$.