Where is the root in these words: miniature, minimal, minimize?

It turns out there are two unrelated roots:

There is the (modern) Portuguese/Galician river Minho/Miño, called Minius in Latin, probably from an unknown root in Iberian, which is assumed to be a language isolate, i.e. not at all related to surrounding languages. From this comes miniature and the modern English prefix mini-.

From classical Latin Minius "an Iberian river":

• → minium "cinnabar" (probably obtained from or near the river Minius)
  • → Postclass. Lat. minio "to colour something (red) with cinnabar, illustrate"
    • → miniatura "a colouring, illustration (typically small)"
      • → Eng. miniature "a small version of something".
        • → mini- "very small"

Then there is Proto-Indo-European *m(e)i- "small" + suffix *-nu (probably a verbal suffix):

• → PIE *minu- "to lessen, reduce*
  • → Lat. minus "less", minor "less, smaller", minimus "least, smallest", minister "servant"
    • → Eng. minus, minor, minuscule, minimize, minimal, minister
  • → Lat. (di)minuo "lessen, decrease"
    • → Eng. diminish, minute, mince

In all these English words, only -ize and -al would seem able to produce new words in English; the rest are best considered purely Latin suffixes. English words using the Latin suffixes -tus, -or, -us, -(i)mus, and -ter were generally taken from Latin (mostly through French), not formed in English.

The modern prefix mini- seems to come from the word miniatura, but the etymology keeps going back, linguistically speaking. Read all the answer, when you reach the end, it will take you back to this first part so you'll understand why "it comes from miniatura".

Miniatura comes from the Latin word miniare, which comes from minio (we can stop here, but it still goes back. I'll mention it if you're interested).

The Minio¹, in chemistry, is the mixture of 2 molecules of monoxide and 1 molecule of lead bioxide, red-orange coloured, obtained from the massicot. This substance was used in Middle Ages in the "Art of miniatura".

The miniatura was the image painted, with extreme care for detail, to decorate the first letters of manuscripts, something like this:

In the italian wikipedia for "miniature", it says (I'll translate it):

Starting from the XIV century, the spreading of such images of small dimension, contributed to make the word "miniature" indicate images, paintings and objects of small dimensions.

(1): The name comes from a river in Spain, called Minius.

Minĭmu(m) ("piccolissimo" = very small) is the superlative for Mĭnus ("piccolo" = small).

Minority and minus come from the Latin word minor, which means "smaller" or "less".

Minute comes from minutus, past participle of mĭnŭĕre, "to make small, to reduce, to shrink".

We can't consider "-ute" "-im" as separated suffixes, as you proposed, because those parts come from the original word.

I couldn't find any proof that all those words relate to each other, and at this point, I could only speculate. My speculation was that "minio" for miniature was the ultimate original word, but during my research, unexpectedly, Greek came into the show and this makes me doubt about my speculation. I'm sorry I can't help you further, hope this helps though. If something comes up, I'll make sure to tell you.