I had an SV650 that used to do that in cold weather .. 
It also suffered horribly from Carb Icing as the Carb Heaters had packed up
Just realised if it is manual choke it is carbs.
For the Carbs to ice up it would need to do a few miles first - running just fine before they ice up? that has been my experience of icing carbs.
But perhaps it is normal going by your post for Suzuki's to do this?
If it is not normal then I suspect it is this
If an engine is running on choke then you are enriching the air / fuel ratio yourself with the choke in other words not enough fuel or too much air is getting into the system. In other words the engine is running lean.
Check the following
1. An air leak has developed around the carburetor areas and is allowing too much air into the combustion chamber.
Check the seals and rubbers around the Carbs for leaks.
2. Fuel system contamination has resulted in the air and fuel regulator jets inside of the carburetor to become clogged.
I do not know your model but inside the Carbs should be an emulsifier tube that allows air and fuel to mix to help the fuel to atomise when it is "injected" into the inlet track.
3. Fuel jets are blocked.
This would mean not enough fuel is getting into the system.
4. Often carbs can also play up if they are not in synchronised.
So synchronising the Carbs could be a good idea.
5. Fuel line is not pinched etc and you have a fuel restriction.
Check your fuel line - if you have a fuel pump I would think that is OK cause your bike does run fine after a while it just struggles while it is cold.
I.e. give your carbs a really good service and clean.
If your engine is running lean this can cause damage - it is better for an engine to run rich than to run lean - so check it out if I was you and not assume it is normal. {Which of cause it could be, it is cold and Carbs can be a little temperamental}
