You can either carry the other yarn across the back - just catch it with the other yarn every few stitches, or use another ball of the brown at the other end, just twist the yarns round each other once so you don’t get a hole (there are videos which illustrate these things in case you need to see!). Using another brown ball makes more sense if you don’t need a thicker fabric or want to save yarn.
As this is continued for a few rows (e.g. next row is the same but purl stitch) how do I make sure that the caught yarn that’s carried across the back is always on the ‘wrong’ side?
As long as you do not take it between the knitting needles it will stay on the wrong side. After twisting with the working yarn you might have to give the brown yarn a slight tug if you notice it is visible from the front side.