Synaptic is normally excellent and it' really odd you're getting a problem.
The way to see if it is indeed runing, is as follows;
1. Open a terminal
2. At the prompt type:
and hit Enter.
(all commands have to be executed with the Enter key)
3. This will tell you a number of things, but the items we are interested in are the second column (PID) and the last (Command).
4. Look down the commands and see if there is one for Thunderbird. If there is, note the PID that relates to it. If there's more than one, note all the numbers associated with Thunderbird and simply repeat the following for each PID.
Now go back to the command prompt and type this command:
sudo kill -9 [PID]
As you're demanding a executive command, you'll be asked for your password. Type it in and hit Enter again.
For example, if the process identification (PID) associated with Thunderbird was 3916, the command would be:
sudo kill -9 3196
The -9 simply ensures execution, in other words, you are telling the system to kill that process no matter what.
See what effect that has and come back if you need further help.