there are two things to look at in your case. 1) constructions that use an infinitive and 2) use of subordinated sentences.

1) When there is a conjugated verb in the sentence, the infinitive can sometimes be used a) as a noun

Rire est le propre de l’homme

b) when you use two verbs in a row

Je voudrais vous parler

or c) introduced by a preposition

Je promets de ne rien dire

2) you can continue a phrase with another subordinate phrase including a second conjugated verb. Just make sure that you follow the “concordance des temps” in this case. The fact that you use (or not) infinitives in the main or in the subordinated sentence (or in both) doesn’t really matter. You must focus on the conjugated verbs

That being said, your two examples are grammatically correct, but the use of “fera tourner” in the second sentence is not appropriate. I would rather say something like

Nous voudrions croire que notre voiture roulera encore plusieurs années.

Remark : a special case of b) is the use of “aller(conjugated) + infinitive, which is used to express an action in the near future (similar to “I am going to do something”).

Your sentence “Nous voudrions croire que notre voiture fera tourner fort plusieurs années encore” is unfortunately not correct, because “fera tourner fort” makes no sense in French. You might have said:

Nous voudrions croire que notre voiture pourra rouler plusieurs années encore

Here, we use the infinitive just like you can do it in English “will be able to roll“

The infinitive might seem a little complicated in the case of the verb “faire” because it is particularly used as a double infinitive, even sometimes with itself:

Je prends le chien, je vais lui faire faire un tour

This means: “I take the dog, I’m going to give it a walk” (“faire un tour” is an idiomatic expression meaning “to take a walk“)

Another example of double infinitive:

Si le voleur fait du bruit, il va se faire prendre .
This means: “If the burglar makes noise, he will get caught“

I can understand the difficulty you feel, because today, even many French people don’t really understand the double infinitive, mainly because of the phonetical identity between infinitive and past participle in the verbs of the “premier groupe” (first group). It is more and more frequent to see mistakes of the following type:

• wrong: Il s’est fait attrapé or il va se faire attrapé (wrong as well)

• instead of: il s’est fait attraper or il va se faire attraper (correct)

Apart from “faire” there is no real difficulty in comparison with English, except for the past conditional, where English doesn’t use the infinitive:

Present: “Vous devez essayer cela”, in English: “You have to try that”: this is quite similar, both languages use the same structure, including the infinitive (“essayer” = “to try”).

But in the past conditional, things get different:
Past: “Vous auriez dû lui dire”, in English: “You should have told him”.
You can see that in the past conditional, French stil uses the infinitive, meanwhile English doesn’t.

I think that this conditional past, apart from “faire”, must be the only difference between both languages, except for the fact that, in French, the infinitive can be used as a noun:

Leur condition leur assurait le boire et le manger

Here, “boire” et “manger” are infinitive verbs taken as nouns (“drink” and “food”).