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:
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").