It does not really make sense to directly compare the value of the derivative with the value of the function. It is like comparing how fast a car is travelling with the position of the car on the road. If you really want to compare somehow, it makes more sense to compare the speed of the car, which is the distance travelled by the car per some unit of time, with the distance between two points on the road, that is how far the car has travelled (not the position of the car). Note that we must agree on the unit of time to measure speed, as in km/hour or km/sec, otherwise just by changing the unit of time you can make the numerical value of the speed as large as you want, making the comparison meaningless.
In formulas, what I am saying is that $f'(x)$ is the derivative of $g(x)=f(x)+C$ for any constant $C$, so by choosing $C$ very large positive or very large negative, you can get any of the two inequalities $g'(x)<g(x)$ and $g'(x)>g(x)$ for any fixed $x$. But the difference $g(x)-g(y)$ (for any two points $x$ and $y$) stays the same no matter what the constant $C$ is, so it is a more meaningful quantity to be compared with $g'$. This is still not enough, because if we consider the new function $h(x)=g(ax)$, then $h(x)-h(y)=g(ax)-g(ay)$ and $h'(x)=ag'(ax)$, so that by choosing $a$ very large or very small, one can make any of the two inequalities $|h'(x)|<|h(x)-h(y)|$ and $|h'(x)|>|h(x)-h(y)|$ true. A way to fix this would be to require $x-y=1$, or to consider functions that are defined on the interval $[a,b]$, etc. I think you know what I am doing here: It introduces a "standard length" in the $x$-direction, which is like introducing a unit of time in the car example.
In any case, assuming that we choose one of the legitimate ways of comparing, if you were blindfolded and to pick a function from a zoo of functions, there is a very large chance that the derivative of the function will be much much larger than the function. This is because a function can oscillate very fast even though having a very limited range of values. An example is $\varepsilon\sin(nx/\varepsilon)$ with small $\varepsilon$ and large $n$. Another crude explanation is that you are allowed to divide by a number that is as small as you like in the definition of derivative, but you are dividing by 1 or something moderately sized in the other expression $g(x)-g(y)$.