Why is the Turkish president's surname is spelt in English as Erdogan, with g?
Solution 1:
Besides the visual similarity between the letters ğ and g, is there any other reason why Erdoğan and many other Turkish proper names are spelt in English with the letter g?
The visual (and historical) relationship between ğ and g is the most important factor. There isn't a big tradition in English of spelling names from other languages according to an English-based transcription of the names' pronunciations. Instead, names tend to have a spelling based on:
The original spelling (if the source language is written in the Latin alphabet); or...
If the source language is not written in the Latin alphabet, some existing transliteration or transcription that is viewed as a "standard" (not necessarily one designed to show pronunciation to monolingual English speakers). E.g. the pinyin transcription of Chinese has become more or less standard, even though it uses letters like X and Q in ways that are unknown in the spelling system used for English words.
Sometimes there are older spellings that don't follow one of these two patterns, but they tend to become deprecated over time. E.g. the spelling "Hindoo" has now been replaced more or less entirely by "Hindu"; the spelling "Koran" has been overtaken by the spelling "Qur'an".
Removing "funny foreign lines, dots and squiggles" from letters or words is another habit of English speakers. Hence, Erdoğan > Erdogan. English speakers don't instinctively think of ğ as a separate letter from g; they think of it as a g with something on top. The use of diacritics in English texts varies a lot and is discussed in other posts on this site, so I won't try to cover it here. It is not impossible to encounter the spelling "Erdoğan" in an English text.