Rear tire feels unstable in turns
 

Manual states 110/90-16 59P for the front 130/90-15M/C 66P for the rear. Also states: The use of tires other than those specified may cause instability. It is highly recommended to use the specified tires. I have always gone with what the manual suggest. Perhaps there are others who have not and can advise if they had any issues.

The 100/90 on the front will work o.k.

Ahem...

Modifying the GZ - What's Available?
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There aren't many tire options for our bike that come in a matching set. There are, however, many options if you don't mind mixing up tire brands and/or going to a slightly smaller tire up front.

The factory tires sizes are: 110/90-16 (F) & 130/90-15 (R)

The rim width for the front wheel is 2.5" and the rear wheel width is 3.0"
That being the case, a size 100/90-16 front tire will work on this bike. It does have an overall diameter decrease of 9mm over the 110/90-16 but that isn't really very drastic.

Samato has verified that alternate tire size 100/90-16 will mount up with no problem.

As well, the 120/80-16 alternate tire size is another choice for the front tire and has been successfully mounted by MRLMD1.

Factory replacement tires: 110/90-16 & 130/90-15
Dunlop D404 (only available through a Suzuki dealer)
Metzeler ME77 (only available through a Suzuki dealer)
Cheng Shin C906 & C907 (possibly no longer available)

Rear Factory Replacements:
Kenda Challenger
Pirelli MT66 (Route 66)
Maxxis M6103
Cheng Shin Marquis
Metzeler Me 77

Factory Replacement Front:
GBC Green Shark
Metzler Lasertec
Pirelli Sport Demon
Avon Roadrider AM26
Conti Ultra TKV11
Kenda K671 & K673

100/90-16 Front:
Bridgestone BT45
Maxxis M6002
Cheng Shin C6000
Kenda Cruiser
IRC Durotour

Matching tires with low profile front: 100/91-16 & 130/90-15
Kenda Cruiser

It could also be alignment. A quick check for alignment is to drive with your hands off the steering (where appropriate, if your comfort level allows, etc.). If you have to lean left or right to keep the bike going straight, your alignment is off.

Just to expand on the above statement. Alignment could be front or rear of the bike. The rear due to not doing a proper chain adjustment and the rear adjustment marks are not the same distance on both sides of the swing arm. This will have the rear tire slightly cocked to one side and it will not follow the front in the same path down the road. This will also accelerate chain and sprocket wear too.
The front could be out of alignment because the forks are slightly tweaked and will need to be adjusted to match each other. If the bars are not totally aligned with the front wheel, such as being slightly left or right when riding in a straight line you will need to investigate further.

Can you check your rear wheel by making some DIY rear fork stands and elevating the rear tire off the ground. (There is a link to making these on the maintenance page)

Once elevated, just start the bike a let the engine momentum turn the rear wheel while you watch it. Check the alingment marks on the rear axle nut, and check the rear axle nut for proper torque. These are all covered somewhat in the front sprocket replacement guide as well.