Translation:I go slowly on narrow roads.
In my point of view, going slowly on a specific road would have a different nuance from what is stated by duo. The sentence meant something like " As for narrow roads (every narrow road), i go slowly on them". It is not wrong, but feels out of place if you want to be specific.
For me it helps to keep them connected with their opposites in my mind: hosoi/futoi and usui/atsui - you can see that they have similar endings based on the pairing, and since I learned futoi as "fat" first, remembering just "futoi = fat = thick in a round/cylindrical way" helps me to figure out the rest, so "hosoi = opposite of fat = thin in a cylindrical way", which means the "ui" ending ones must be the "thin/thick in a flat way" meanings.
ほそい = 'thin' for roughly cylindrical objects (i.e. ones that use the counter 本), also translatable as "slender/fine" and - in the case of roads - "narrow".
うすい = 'thin' for roughly flat objects, such as books, boards, slices of bread etc. It also means 'weak' when you're talking about coffee/tea etc., as well as 'light' when speaking of colo(u)rs.
進む（すすむ）is not something you do 'in/on' a certain location. It's the act of progressing/advancing along, and usually the location thereof actually takes the particle を, just like verbs for "to walk", "to cross" or "to turn [a corner]". は could be used here in an implied juxtaposition to wider roads in general, or in reference to a previously mentioned road.