As also mentioned by kspalla, cascading a network with low loss but reasonable VSWR (your balun) with a load with reasonable VSWR (your antenna), can result in bad overall VSWR. So you should optimize for better VSWR for both antenna and Balun.
There is more that goes wrong: around 3.25 GHz, RL(antenna) = 9dB, RL(balun) = 25 dB, RL(ant+balun) = 4dB
A lossless network with RL=25 dB and a load with RL=9dB, would result in a worst case RL=7.9 dB, though your simulation shows 4 dB.
Did you simulate the balun together with the antenna in one EM simulation (so not a simulation based on S-parameter data)? If so, check for possible electromagnetic interference between the balun and the antenna itself. In other words, the field of the antenna may interact with the balun and that will change the antenna's impedance (and radiation pattern). This does not solve your problem as RL of both antenna and balun is not good enough.
If you did overall simulation based on S-parameters (this assumes no electromagnetic interaction), something is wrong with your reference impedance used during the EM simulation.