How and what, exactly, have you insulated, and where is your domicile in relation to the street?
Do you have sound dampening mats on the ceiling? If not, bear in mind that most houses and apartments are above street level, and most of the sound will be reflected off the ceiling. A layer of sound dampening material there should have the largest effect.
If you live low to the ground, sound insulating the walls that can see the street, rather than just outside walls would have a similar effect.
There are two components to the noise intrusion - the direct field and the reverberant field. Adding absorptive finishes to the room will help reduce the reverberant field, which in a best case scenario will buy you 3dB (i.e a barely noticeable improvement). Short improving the isolation (i.e windows and exterior partition construction) there isn't anything you can do about the direct field.
A few strategically placed plants or sound dividers - think cubicle walls but far less intrusive - can also help.
Massive barriers can help in outdoor noise propagation scenarios. In this type of situation, the outdoor noise is impacting the exterior facade, causing it to vibrate, and is re-radiating sound in the interior of the OP's space. This means that the source of sound is a large area, not a point source. The sound is effectively coming from "everywhere", and so you don't get the same kind of path length difference attenuation from barriers like you do when you have a point source that is far from the receiver. This is exascerbated by the fact that the intrusion is low frequency which diffracts around corners far more than higher frequency sound.
Plants are not massive enough to be of any significant help. To put things into perspective, a dense forest that's 100m in depth will only attenuate sound 2-3dB, and that's mostly a high frequency reduction.
If you must go with a noise generator (which I don't recommend), try pink noise instead of white. The sounds from the street you try to mask out are going to be mostly low frequency, and white noise will mainly add more sound energy in the higher end of the spectrum.
White noise will add equal sound energy across the spectrum. Pink noise will add more energy in the lower frequencies. We don't typically recommend noise masking for low frequency intrusions into offices and other facilities we consult on because the masking is generally perceived to be more annoying than the original problem. It's best used when there is a speech privacy problem in large open offices because there isn't ENOUGH background noise.
Upgrading your window would typically be recommended, but will only do a little for low frequency. Note that in glazed assemblies, the framing is the weak point. STC 35 is where most frames will top out (even though the glazing will advertise higher STCs into the 50s). If you want a really solid window construction you need an isolated frame assembly which is two frames in series that do not touch each other and which are isolated from the surrounding structure using 3mm thick neoprene gaskets. Each frame supports a separate pane of glass. The inside perimeter should be lined with glass-fibre to prevent standing waves in the cavity.
Overall thickness of the glazing might be 2-3"