As a rule of thumb, the prefix ex- is pronounced with /ks/ when the prefix is stressed.:

• 'excellent
• 'exit
• 'exile
• 'execute

When this prefix is not stressed, then if the first sound in the root (the part after the prefix) begins with a voiced sound, the prefix will be pronounced with /gz/:

• ex'am
• ex'asperate
• ex'actly
• ex'ist

This is still true, of course if the prefix is followed by a silent /h/ in the writing. In this case of course, the first sound in the root is a vowel sound:

• ex'hibit
• ex'haust
• ex'hort
• ex'hilaration

If the prefix is followed by an unvoiced sound, then it will be pronounced with /ks/

• ex'pect /ɪkˈspekt/
• ex'tol /ɪkˈstəʊl/
• ex'hale /eks'heɪl/
• ex'ceed /ɪkˈsiːd/
• ex'foliate /eksˈfəʊlieɪt/

In the words above we see roots beginning with the following voiceless sounds respectively /p, t, h, s, f/ and so the prefix used /ks/ instead of /gz/.

There is a different prefix in English exo- meaning 'outer' or 'outside'. The meaning is similar to ex- which tends to mean 'out of' or 'from'. The prefix exo- is always pronounced with /ks/. It is a Greek prefix. Another Greek prefix, hex(a), meaning 'six', is also pronounced with /ks/:

• hexagon
• hexadecimal

X at the end of a word nearly always represents the sounds /ks/.

• latex
• six
• Bronx

Very few words in English begin with X. Those that do are mostly from Greek and tend to be pronounced with /z/:

• xylophone
• xanthar gum
• xenon

TLDR

The pronunciation of 'iron' in standard varieties of English is EYE-URN (BrE: /'aɪən/, AmE: /'aɪrn/) and not EYE-RUN (which is also a common pronunciation of 'iron' in some varieties of English) because of a very common process called Metathesis. It's defined as the transposition/rearrangement of letters, syllables or phonemes (sounds) in a word.

Explanation

There are some other words that show the same change; 'wasp' used to be waps, 'bird' used to be brid and 'horse' used to be hros, but they've changed over time. Why is that?

It's because of a very common process called Metathesis. /'aɪən/ is the metathesised version of (/'aɪrən/). 'Iron' commonly used to be pronounced the way it's spelt (/'aɪrən/), but due to metathesis, its pronunciation became EYE-URN (/'aɪən/). Other words such as horse, bird, third etc., reflect the change in spelling; however, 'iron' doesn't reflect that change probably because metathesis applied to it after the spelling was standardised.

Metathesis:

Metathesis is defined as ’the transposition/rearrangement of letters, syllables or phonemes (sounds) in a word’.

Examples:

• 'wasp' used to be waps (wæps) [transposition of p and s]
• 'bird' used to be brid [transposition of i and r]

{Historical Metathesis}

• 'mix' being pronounced /mɪsk/ rather than /mɪks/ [transposition of k and s]

• 'desk' being pronounced /dɛks/ rather than /dɛsk/ [transposition of k and s]

• 'modern' being pronounced /'mɒdɹən/ instead of /ˈmɒd(ə)n/ (US: /ˈmɑː.dɚn/)

• 'pattern' being pronounced /pætɹən/ instead of /ˈpæt.ən/ (US: /ˈpæɾ.ɚn/)

• Another famous example from Shakespeare's 'The Tempest' is the figure of Caliban whose name originates from a phonological metathesis of /n/ and /l/ in 'cannibal'. _[ThoughtCo]

/r/-metathesis

Words having /r/ + vowel sequences are more susceptible to metathesis than others. According to A grammar of Old English Phonology by Richard Hogg, ‘R-metathesis normally occurs when /r/ is followed by a short vowel and a dental or alveolar consonant, usually /n/ or /s/’.

'Iron' is an example of /r/-metathesis. It was probably pronounced /'aɪrən/ (EYE-RUN) at one point, but it got metathesised to /'aɪərn/ (EYE-URN). However, the spelling remained unaffected.

Other examples of metathesis of /r/ include:

• bird from brid (bridde)
• third from thridde (ðridde)
• horse from hros

The silent R in 'iron' in BrE

The reason why the r in 'iron' is absent in British English is because the r is followed by a consonant now (followed by /n/ in /'aɪərn/) and British English is non-rhotic, meaning the r is only pronounced when followed by a vowel. The same thing happened to 'bird', 'horse' and 'third' too (i.e. the r is followed by a consonant, so it's silent).

There are different types of metathesis, 'colonel' (pronounced KE(R)-NUHL /'kɜː(r)nl̩/) can also be said to be a product of metathesis. (See this answer on ELU for the spelling and pronunciation of 'colonel')

According to Wikipedia, the reason for ‘common speech errors’ is also metathesis.
Examples include:

• perscription for prescription
• interduce for introduce
• revelant for relevant
• foilage and foliage

References:

• A Dictionary of Phonetics and Phonology - Trask and Lawrence
• English Words: History and Structure by Stockwell and Minkova
• A Grammar of Old English Phonology by Richard Hogg
• Metathesis - ThoughtCo
• The Sounds of Language: An Introduction to Phonetics and Phonology - Elizabeth C. Zsiga