I am trying to find out if there is a convention of correctness for writing Arabic proper nouns starting with 'Al'

The news channel 'Al Jazeera' writes its name like I have i.e with a space between 'Al' and 'Jazeera', in text but in the logo it is 'ALJAZEERA'. One come across variations like 'Alqaeda', 'Al-Qaeda', 'Al Qaeda', and even 'al-Qaeda'. I am trying to find out if there is a proper convention for writing Arabic proper nouns starting with 'Al'.

This is a style question—and one that different style guides have addressed in markedly different ways.

The Chicago Manual of Style, sixteenth edition (2010) expresses clear preferences both with regard to hyphenating al and with regard to lowercasing it:

11.99 The Arabic definite article. Though there is considerable variation across publications, Chicago recommends joining the Arabic definite article, al, to a noun with a hyphen.

[Examples:] al-Islam, al-Nafud, Bahr al-Sufi, al-Qaeda (or al-Qaida)

...

11.100 Arabic capitalization. Since the Arabic alphabet does not distinguish between capital and lowercase letter forms, practice in capitalizing transliterated Arabic varies widely. Chicago recommends the practice outline in 11.3: capitalize only the first word and any proper nouns. This practice applies to titles of works as well as to names of journals and organizations. Note that al, like the, is capitalized only at the beginning of a sentence or title.

[Example:] 'Abd al-Rahman al-Jabarti, 'Aja'ib al-athar fi al-tarajim wa al-akhbar (The marvelous remains in biography and history)

Refreshingly, the advice in The Oxford Guide to Style (2002) is almost identical to that in Chicago:

11.6.2 Transliteration

...

The definite article al should always be joined to the noun with a hyphen: al-Islam, al-kitab. ...

...

11.6.2.2 Capitalization

Problems of Arabic capitalization occur only in transliteration, since the Arabic alphabet does not distinguish between upper- and lower-case letters. The definite article al normally is capitalized only at the beginning of a sentence, but not at the start of a reference title.

So both Chicago and Oxford recommend the forms al-Jazeera and al-Qaeda/al-Qaida, unless the term appears at the beginning of a sentence.

Meanwhile, The Associated Press Stylebook (2007) blithely embraces inconsistency, without explanation, as it offers the following entries:

Al-Jazeera

...

al-Qaida International terrorist organization headed by Osama bin Laden.

The New York Times Manual of Style and Usage (1999) departs from AP style in yet another idiosyncratic way, as this entry suggests:

Al Fatah, the Palestinian guerilla organization. Omit Al when the name is preceded by a or the: a Fatah leader. In headlines: Fatah

I have noticed that the New York Times still adheres to this general rule in it treatment of the name Al Qaeda, leading to headlines like this one from December 6, 2014:

Pakistani Military Kills a Qaeda Leader

In my view, the Oxford/Chicago approach is the most consistent and logical, but as long as you adhere to one approach instead of mixing and matching, you can generally avoid distracting or baffling your readers.