"I'll make dinner" ("I will make dinner" ) is future tense, and therefore would have a different conjugation of 'hacer' - "Yo haré la cena". 'Hago' is present tense meaning "I make".

why?

I was surprised, too. I thought this would be OK.

Yo hago. Él/ella/usted hace.

Hacer is an irregular verb. It's not going to conjugate the way many other -ER verbs do in the present tense.

I wrote ''I make the dinner'' and it was correct

"Hago" is the 1st person singular, present tense form of "hacer".
Yo hago = I make/do/etc.
Tú haces = you make/...
Él/Ella/Usted hace = he/she/you(formal) make
Nosotros hacemos = we make
Vosotros hacéis = you all(formal) make
Ellos/Ellas/Ustedes hacen = they/you all make
The 1st person singular form has an irregular conjunction.

yes!

Present tense forms of "hacer":
Yo hago
Tú haces
Él/Ella/Usted hace
Nosotros hacemos
Vosotros hacéis
Ellos/Ellas/Ustedes hacen

I would guess because adding "do" is an English way of adding emphasis or improper grammar. If the latter, then it's incorrect English. If the former, then there is probably a particular way of adding emphasis in Spanish that has not been covered.

'Hago' (I make) is the present tense of 'hacer' (to make/do/etc.) and 'hice' (I made) is the past tense of 'hacer'.

"Hago" (1st person singular form of "hacer") means both depending on the context. "Hacer" has a variety of meanings.

The dictionary in the website Reverso is a great resource to learn the meaning of Spanish (or French, German, etc.) words, and the conjugator is a fabulous tool to learn all the different conjugations of verbs.
The link below lets you know all the meanings of and ways to use "hacer" (AKA hago).
http://diccionario.reverso.net/espanol-ingles/hacer