It sounds like the GM wants the spell long arm to apply to your PC's limbs the weapon quality reach rather than increasing your PC's natural reach to like that of the typical creature that possesses the size category Large (tall).
While the GM is free to rule any way he wants, the spell long arm simply says, "Your arms temporarily grow in length, increasing your reach with those limbs by 5 feet." The spell does not say, "Your arms temporarily grow in length so that weapons you wield with them that lack the weapon quality reach gain the weapon quality reach"—this would not only be extremely complicated but also little strange: I mean, generally, if that were the effect you wanted, you'd just wield (or even draw!) a weapon that had the weapon quality reach instead!
Convincing the GM that the spell long arm actually means natural reach may be difficult because the game has two entirely different definitions of reach, one for weapons that are particularly long and don't usually threaten adjacent squares, and another for a creature's natural reach that allows it to threaten all the squares around it. I urge showing the GM the diagram linked to above for a Large (tall) creature then this Order of the Stick comic strip, and, if those fail, having your PC (armed with a non-reach weapon) get all up in every ogre's grill—where, if the DM's consistent, your PC should be totally safe—until the GM changes his mind.