In Old English it was already pronounced /j/ there, even though the spelling was still "weg". Ditto for Old Frisian, so Normans had nothing to do with it.

The process is actually fairly straightforward. First you start with a common language that has two allophones for /g/ which are ~ [g] and [ɣ], depending on context; in this case, "weg" was [weɣ].

In Old Dutch [ɣ] then becomes subject to final obstruent devoicing, giving [x] of the modern Dutch pronunciation of the same word.

Meanwhile in Old English /k/ and /g/ (in either of its incarnations) palatalized in various environments instead. For [ɣ] in particular, it became palatalized after [e] in most cases - thus we get [weʝ]. And then [ʝ] is already very similar to [j], and gradually evolved into the latter. This all has already happened by the time most Old English texts were written.

something in between obviously ghjesterdaj

The "y" in "day" went through the same process, so in your theory it would be something like "ghjesterdaghj". Or maybe it was still just "ghjester" back then, dunno. Unfortunately, I don't think we will ever know exactly what it sounded like.