First: Ibn Sina and Al-Khwarezmi were Persian. This distinction is very important, do not conflate "Islamic" with "Arab"!

Between any languages, there are, in addition to different consonants, consonant clusters that are more or less common. Uncommon consonant clusters are difficult for non-native speakers to pronounce, and hence historically and sometimes today tend to be modified in addition to replacing non-native consonants.

The names you mentioned would all be written in Arabic/Perso-Arabic script, which introduces some additional arbitrarines in rendering them in Latin script. So let's look:

Ibn Sina - "Ibn" has the b-n cluster, which is uncommon in Romance languages; hence it becomes "Avi".

Ibn Rushd - Same thing as above, with the addition of "sh-d" - not so weird in English but not easily rendered in Latin.

Al-Khwarezmi - "Kh" is here a "soft k" roughly as in loch or German ich. This consonant is also alien to Latin. One might expect -zmi to become -smi rather than -tmi - I can't explain why this happened exactly.

While today "authentic" and consistently transliterated rendering of names can be important to cultural sensitivity and even diplomacy, back in the day, people would be happy to modify a foreign name to fit their native tongue. This isn't limited to medieval Latin - the Achaemenid Great King Khshayarsha became Xerxes in Greek; Kourosh became Kyros, Zarathustra became Zoroaster (some folk etymology played a role here - Zarathustra means "loved by camels", cf Phillippos "loved by horses", but the Greeks interpreted the name to mean "undiluted" zoro- + "star" astra). Of course, we still do this to an extent when speaking by necessity - most English speakers cannot pronounce e.g. the names Khomeini or Khamene'i correctly, but substitute "k" for "kh".