Врач сказал, что сначала нужно сделать рентген, чтобы исключить перелом.

Because the sentence reports what the doctor said.
Врач сказал = The doctor said.

Russian uses the past tense here the same way English does when introducing reported speech:

• Врач сказал, что... = The doctor said that...

Also, сказал is masculine singular, so it agrees with врач, which is a masculine noun.

Because что introduces a subordinate clause.

The sentence has two parts:

• Врач сказал = main clause
• что сначала нужно сделать рентген... = subordinate clause

In Russian, subordinate clauses are normally separated by commas, so:

• Врач сказал, что...

Here что means that.

So:

• Врач сказал, что... = The doctor said that...

In Russian, что is very commonly used to introduce reported statements:

• Он сказал, что придёт. = He said that he would come.

Сначала means first / at first / first of all.

In this sentence it modifies the whole idea of what needs to be done:

• что сначала нужно сделать рентген
  = that first an X-ray needs to be done

Russian word order is flexible, but this placement is very natural because it highlights the sequence:

1. first do the X-ray
2. then decide based on the result

You could move it in some contexts, but сначала нужно... is the most natural phrasing here.

Нужно is an impersonal word meaning it is necessary / need to / one must.

So:

• нужно сделать рентген = it is necessary to do an X-ray / we need to do an X-ray

This is different from должен, which agrees with a specific person:

• Я должен = I must
• Он должен = he must

Here the sentence does not say exactly who must do it; it gives a general medical necessity. That is why нужно sounds natural.

Because сделать is perfective, and here the speaker means a single completed action: to get / do an X-ray.

Compare:

• делать = to do, to be doing, to do repeatedly or in general
• сделать = to do and complete

In this context, the doctor means one concrete action that should be completed:

• нужно сделать рентген = an X-ray needs to be done

If you said нужно делать рентген, it could sound more like a repeated or ongoing action, which is not the intended meaning here.

In everyday Russian, рентген often works as a shortened way to mean an X-ray or an X-ray examination.

So сделать рентген is a normal, common phrase in speech.

More formal or more specific versions include:

• сделать рентгеновский снимок = to take an X-ray image
• сделать рентген = to get/have an X-ray

Native speakers often prefer the shorter version in conversation.

Чтобы means in order to / so that.

When the subject of both actions is general or the same, Russian often uses:

• чтобы + infinitive

Here:

• нужно сделать рентген, чтобы исключить перелом
• an X-ray is needed in order to rule out a fracture

This is very natural after words like нужно, надо, можно, etc.

If Russian wanted to express a full clause with its own subject, it could use a finite verb after чтобы, but that is not necessary here.

Because the goal is a completed result: to rule out a fracture.

Исключить is perfective and means to exclude something successfully as a possibility.

Compare:

• исключать = to be excluding / to exclude in general
• исключить = to exclude, to rule out completely

After чтобы, Russian often uses the perfective infinitive when the purpose is to achieve a definite result.

Because перелом is the direct object of исключить:

• исключить что? → перелом

For masculine inanimate nouns in Russian, the accusative singular is often the same as the nominative singular.

So:

• nominative: перелом
• accusative: перелом

That is why its form does not change here.

Not necessarily.

Врач is grammatically masculine, so past-tense verbs normally appear in the masculine form unless the speaker specifically marks the doctor as female:

• Врач сказал = the doctor said
• Врач сказала = the doctor said, if the doctor is female

In real life, врач can refer to a doctor of either sex. Russian often uses the masculine noun as the default profession word.

Usually no, not in standard Russian.

English can say:

• The doctor said that...
• The doctor said...

But Russian normally keeps что:

• Врач сказал, что...

Leaving it out would sound unnatural or incorrect in ordinary standard speech.

No. They express different relationships.

• чтобы = in order to / so that → purpose
• потому что = because → reason

In this sentence:

• сделать рентген, чтобы исключить перелом
• the X-ray is done in order to rule out a fracture

That is purpose, not cause.

Yes, надо would also be possible:

• Врач сказал, что сначала надо сделать рентген...

Both надо and нужно can mean need to / it is necessary to.

A rough nuance:

• надо is a bit more conversational
• нужно can sound a little more neutral or formal

In this sentence, either works well.

A helpful breakdown is:

• Врач сказал
  main clause: The doctor said

• что сначала нужно сделать рентген
  content clause: that first an X-ray needs to be done

• чтобы исключить перелом
  purpose clause: in order to rule out a fracture

So the full structure is:

[main clause] + [that-clause] + [purpose clause]

This is a very common Russian sentence pattern.