“Je passe un examen” is a correct form. “je suis passer…” is wrong.

You can say “je vais passer un examen” to mean “I’m going to take an exam”
or “j’ai passé un examen” to mean “I have taken an exam“.

If you want to use a present simple as in English, you use “passer”, like “Je passe un examen”, “tu passes un examen”,…
But if you want to use a present Be+ing, you should use “être en train de”, like “Je suis en train de passer un examen”, “tu es en train de passer un examen”, etc…
We use “être en train de” to describe an action that is happening.
I hope I have answered !

Be+ing present is often translated with “en train de + verbe” structure, however it is a more heavy construction than it is in English, so it is not used when not necessary. For example:

Que fais-tu ? – Je révise pour mon examen.

Would be translated by:

What are you doing ? – I’m rewiewing for my exam.

And if your mom was calling you in the middle of your exam, you could tell here:

Je suis en plein examen, rappelle-moi plus tard !

or

Je passe un examen, là, c’est pas le moment !

or even

Je suis en train de passer un examen, et toi comment vas-tu ?

French has one present tense only. English has two (present simple and present continuous). Depending on context, the English present and present continuous always translate to the French present tense. There are some exceptions where one might use en train de, but there is no hard and fixed rule for this.

I speak French every Tuesday at school = Je parle français tous les mardis à l’école.

I’m working at IBM this week = Je travaille chez IBM cette semaine.

That said, in French, être en train de would usually translate as the present continuous in English.

Où sont-ils? Réponse: Ils sont en train de faire la lessive. Translation: They’re doing the laundry.

In the example given, I’m taking an exam. The normal translation is: Je passe un examen. Only in some circumstances might one say: Je suis en train de passer un examen. [Presumably, if you are taking an exam, you would be able to talk about it to anyone unless you’re on a cell, which is probably not allowed]. But even in this case, in French, Je passe un examen would be understood by a French speaker as something happening in the present continuous sense of the verb (am taking) in English.

Je passe un examen

Is right. Others mentioned:

Je suis en examen

That is also ok but can also mean I’m being charged with..

I want to add the simpler form:

J’ai un examen

That is the most popular use in spoken French and will be followed by a time indication, for example:

J’ai un examen ce soir (I’m taking an exam tonight)

If you want to say you just finished it

Je viens de passer un examen

If you want to say you’re passing it right now

Je suis en train de passer un examen

If you want to say you are going to take an exam

Je vais passer un examen

French language is usually slightly more exact with timings and less flexible than english.

In french, I am taking a exam = Je prends un examen. The verb prendre means to take. The verb passer means to pass.
Je passe un examen means I pass a exam.
Je prends un examen -I am taking a exam
Je prenderai un examen – I will take an examen
J’ai pris un examen -I took an exam

Je suis fatigue et Je prends un pause.
I am tired and I am taking a break.