GST is a dimer and very often you have only one of the 2 fully translated or a degradation process going on or rare codon related stops.Could be also  a steric hindrance in the folding etc.Nevertheless the dimer will link both versions: GST-peptide x GST  and this you see on the SDS gel, where both species separate.

I guess you mean free GST and peptide (after cleavage) would appear in the same purified fraction. my explanation  is peptide may have a little non-specific affinity to resin and free GST (bound to GSH) lost the affinity to resin (due to a round of GSH elution, even you should have done dialysis). Then the free GST and peptide could not be separated by 2nd round of GSH-agarose purification. Importantly, you should apply other methods (such as size exclusion or ion exchange chromatography) for further separation. Hope the suggestion is useful.

If your peptide has no affinity to glutathione but is still being purified, this tells you that the tag is cleaved after purification. Trace proteases are the obvious answer, but there really isn't enough information to answer the question. The pGEX vector includes a protease cleavage site after GST, but presumably these proteases aren't present in your protocol. Non-specific digestion is also possible. What is your expression system? Do you use an inhibitor? How is your protein stored and for how long before you see multiple bands? Can you post a gel?

firstly if you have two bands gst has a kda of around 26 and you protein as you have said will have a kda of 4 so with tag will be 30kda so try seperating on a 15% gel and look at the weighting of each band if they are equal then you may cleavage/degradation. If the higher band is more promineant then use gelfilration to separate prior to assay. if you have rare codon related stops try overexpression in Rosetta pLysS as this will help over come this issue, also lower temp and shorter expression can alieviate over expression of the gst tag on its own. GST cleavage sites are vector specific thrombin factor Xa and enterokinase (covers ge and millipore vectors) so if you are using any of these prior to binding then retthink your stratergy.