fare .......

it is difficult to use fare in writing mainly because its origin was from Middle English and the original sense of the word (journey, road..etc.) has since been obsolete except in compound form (ie. well-fare, way-farer, sea-faring...etc.). thus you'll find it more difficult to use the word "fare" in a modern day sentence. BUT, if you must, the following will illustrate probable examples....

when used as a noun
example1 : The fare was far worse than anyone can imagine given the notoriety of the whole ordeal.

example2 :It was the usual fancy literary fare with numerous notable speakers and critics all in full participation.

when used as a verb (without an object)
example1 :They fared sumptuously upon the welcoming feast than anyone would have thought possible.

example2 :Her death fared ill with him for he lost all meaning of life upon her tragic departure.

Fare (noun & verb)
As a noun, fare is pretty uniquely used, but it is quite limitary as a verb. As your sample given above: He fared better in the second test.
I think 99% of people would write the same meaning as: He is improving in the second test.
Do you think it is more understandable & acceptable by most people?