Its probably just an early variety.
Yes, this year's briars will bear next year's fruit, and you need to cut off all the branches that bore this year right from the base. To make this easier many people train all the current years new growth to one side, and the following year train new growth to the other side. This means that in the autumn all the briars to be cut off are together, and you don't have the job of sorting and untangling. I found this tip very useful when we had a blackberry.

Oh, I guess I don't have it quite right then - although I did cut the old ones off. Shall I leave them both sides for now or cut the shortest side off ?

I don't bother to separate the old and new canes, just tie the new ones in as they develop. If you have some wires stretched horizontally at, say 2, 3, 4 and 5 feet, the canes can be trained in a serpentine fashion (which reduces the spread). The old canes are easy to spot - they have side shoots which held the fruit - and can be cut out, leaving the new ones intact.

As Andrewr says, I think I probably should have a stout post at both ends, 6 ft tall with wires also as described, not relying on the fence at all. Then I could grow the canes one way.(or both)
But is it too late to disturb what I already have ?

Hank its not worth the bother of changing what you did. Its not at all necessary to do it the way I suggested, just easier when it comes to cutting the old canes off.

O.k.Sue, maybe next year.