Is this correct?

I don´t want to know. = Bilmek istemiyorum.

Is the rule valid in Turkish too that when two verbs come one after the other, the second one is infinitive?

Thanks!

Your translation is correct, but I read istemek (to want) is one of only two Turkish verbs where you can use the infinitive verb beforehand.

Most other verbs need other forms (for example XXXmeyi, XXXmeye) when used together in sentences. 

I like/love to swim (swimming)

Yüzmeyi seviyorum.

I started to swim (swimming)

Yüzmeye başladım.

I want to swim.

Yüzmek istiyorum.

These are all correct:

Bilmek istemiyorum.

Yapmak istemiyorum.

Gitmek istemiyorum.

Uyumak istemiyorum.

Çalışmak istemiyorum.

...

Infinitive marking changes the verb into a noun. A noun is treated like a noun is treated: it takes case endings. Which form is used is not grammar, it is lexical information. We check from the dictionary which case is needed and follow the instructions. For instance we find

sevmek /ı/
1. to love; to like.
2. to fondle, caress. Sevsinler! colloq. Now isn´t he/she something! (said sarcastically). Sev beni, seveyim seni. proverb You scratch my back and I´ll scratch yours.

|ı| means accusative and it shows in the syntax no matter if the complement is a noun or a verb:

                         annem|i seviyorum ACC

                         okuma|y|ı seviyorum ACC

Respectively

                         ev|e gidiyorum DAT

                         yüzme|y|e gidiyorum DAT

Two verbs, istemek and bilmek, however, make an exception. Their demand of accusative governing doesn’t show in the infinitive:

                         elma|yı istiyorum ACC

                         yürümek istiyorum Ø

Anyway, this works only in the simple situations where the one who wants and the one who performs the action are one and the same person. Make the sentence a little bit more complicated (‘I want him to walk’ ) and the ACC ending shows.

I was thinking about the rule that you suggested:

Maybe I would put it this way: there is only one finite verb in a sentence. If there is another verb it has to be subordinated to the finite verb, i. e. marked an infinitive or a participle.

The verb istemek can take either  "infinitive verb" or " verb in accusative form" before it.

Oraya gelmek istiyorum ---> I´d like to come over there. --> Infinitive form

Oraya gelmeyi istiyorum ---> I´d like coming over there. ---> Accusative form

If you like to know the differences between two sentences then look at the "red highlighted" letters. In practice we Turks would use both for the same situation.

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Türkçe öğrenmek istiyorum ---> I´d like to learn Turkish

Türkçe öğrenmeyi istiyorum ---> I´d like learning Turkish.

In both forms the first verb[al] [öğrenmek and öğrenmeyi ]  can be classified as an object of the main verb [istiyorum]

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However, using the accusative case for the verb before´istemek´ is NOT always sounding good in Turkish , such as ;

Çay içmek istiyorum ---> I´d like to drink tea. ---> This is what we would normally say.

Çay içmeyi istiyorum ---> I´d like drinking tea.--> This form is sounding bit weird.

May be it is like in English, you wouldnt say "I´d like [to] drinking tea" instead you normally would say " I´d like to drink tea"

Thank you all very much for the explanations, you´ve helped me a lot.

Maybe it is basically the difference between definite and indefinite object.

I would just choose not to complicate things and as long as the two have the same meaning, use the infinitive.

I want him to walk = Onu yürümeyi istiyorum?

(Onun) yürümesini istiyorum.

yürü|me|si|n|i = verb stem + infinitive marking + possessive suffix sg 3rd + pronominal -n- + accusative ending = ´his walking´