Eski çıkarım yeni kayıtla çürütüldü.

Because Turkish normally puts the finite verb at the end in a neutral sentence.

So the structure here is roughly:

• Eski çıkarım = the old inference
• yeni kayıtla = with / by means of the new record
• çürütüldü = was refuted

This verb-final order is very common in Turkish. Other word orders are possible, but they usually change the emphasis.

Because this sentence is passive.

In an active sentence, eski çıkarım would be the direct object, and if it is definite, it would usually take the accusative:

• Yeni kayıt eski çıkarımı çürüttü.
  The new record refuted the old inference.

In your sentence, the object of that active sentence has become the subject of the passive sentence:

• Eski çıkarım yeni kayıtla çürütüldü.

So it appears in the basic, unmarked form: çıkarım, not çıkarımı.

Çürütüldü can be broken down like this:

• çürüt- = refute, disprove
• -ül- = passive
• -dü = past tense

So:

• çürütmek = to refute
• çürütülmek = to be refuted
• çürütüldü = was refuted

The vowels change according to vowel harmony, which is why you see ü in both suffixes here.

-la / -le means with or by means of. It is the attached form of ile.

So:

• kayıt ile
• kayıtla

both mean essentially the same thing.

In this sentence, yeni kayıtla means something like:

• with the new record
• by means of the new record
• on the basis of the new record

The attached form is very common in everyday Turkish.

Because of vowel harmony.

The noun kayıt contains the back vowel ı, so the suffix appears as -la, not -le.

A few basic patterns:

• back vowels (a, ı, o, u) → -la
• front vowels (e, i, ö, ü) → -le

So:

• kayıtla
• evle would be wrong; it should be evle? Actually for ev, the normal form would be evle only in special combinations, but in ordinary usage you usually say evle only as a suffixal form in limited contexts; with ile, the standard attached form is evle in theory but evle is rare and sounds awkward in isolation, while forms like arabayla are more familiar. The key point for your sentence is simply that kayıt takes -la because of its vowel.

Also, after a vowel, Turkish often uses a buffer y:

• araba → arabayla
• köye is different; that is dative, not this suffix

In English, either translation may work depending on context, but grammatically -la usually marks means, instrument, or basis, not a true agent.

So yeni kayıtla çürütüldü is most naturally understood as:

• it was refuted using the new record
• it was refuted on the basis of the new record

If Turkish wants to express a clear agent in a passive sentence, especially a person or institution, it often uses tarafından:

• Uzman tarafından çürütüldü.
  It was refuted by the expert.

So in your sentence, -la is better understood as with / by means of, not as a human-like doer.

Because Turkish adjectives do not agree with nouns the way adjectives do in many Indo-European languages.

They do not change for:

• gender
• number
• case

So:

• eski çıkarım
• eski çıkarımlar
• yeni kayıt
• yeni kayıtla

The adjective stays the same, and any case or plural marking goes on the noun, not on the adjective.

Turkish has no definite article like English the.

So a bare noun such as çıkarım or kayıt can often be translated as:

• the inference / record
• an inference / record

depending on context.

In this sentence, English naturally uses the because the meaning is specific and known from context. Turkish usually leaves that specificity to context rather than adding a separate word.

The key sounds are:

• ç = like ch in church
• ı = a sound English does not have exactly; it is a close, unrounded vowel. A rough approximation is a very plain, short vowel with relaxed lips.
• ü = like German ü or French u; say ee while rounding your lips

So in this sentence:

• çıkarım starts with ç + ı
• kayıt contains ı
• çürütüldü contains several ü sounds

If those vowels feel difficult at first, that is completely normal for English speakers. The most important thing is to keep ı and i separate, and ü and u separate.