Why can't you have more turning points than the degree?

The problem is that you are confusing real zeros of a polynomial with the degree. These are not the same. The degree of a single variable polynomial is the highest power the polynomial has.

Your hand drawn graph has only 4 real roots, but if it was a polynomial it must have more complex roots. You could not make all those turning points without this been true. You may not be aware of complex numbers.

Although you mention this as precalculus, this does become clearer with calculus, where you find the turning points ( "local maxima and minima" ) by equating the derivative of the polynomial to zero. The derivative of an n-th degree polynomial is an (n-1)th degree polynomial, so their can be as many as (n-1) turning points. However, the derivative's roots need not all be real, and in that case the original polynomial would have fewer real local maxima and minima than n-1.

So the problem is equating the number of real roots with the degree. You can really only know the degree by knowing the highest power the polynomial has. This is not always immediately obvious from the shape of a graph. You also need to be aware of the possibility of complex roots.