I have been a victim of your badly arranged schematic. I thought that the MOSFET was being operated as a high-side switch, which is not the case. Please try better in future to arrange the schematics to show clearly the architecture of the design.
The long answer I wrote below might be moot, but there are potential issues with using a low-side switch for your servo. When "off", the servo's unloaded PWM input will be close to +6V, well outside the MCU's pins' permissble range of potentials 0V to +3.3V. This may or may not be a problem, depending on a few things, including the internal circuitry of the servo itself. In any case, the solution below is still valid, and alleviates the problem.
You can use the N-channel MOSFET in this "source follower" configuration, but to switch it fully on will require \$V_{GATE}=+6V + V_{GS(TH)}\$ to obtain \$V_{SOURCE}=+6V\$. To put this another way, source potential is:
$$ V_{SOURCE} = V_{GATE} - V_{GS(TH)} $$
For the IRLZ44, \$V_{GS(TH)} \approx 1.5\rm V\$, and might even be as high as \$V_{GS(TH)} = 2.0\rm V\$, so your servo is probably only seeing this supply potential:
$$ V_{SOURCE} = +3.3{\rm V} - 1.5{\rm V} = +1.8V $$
You could employ the +9V supply to obtain \$V_{GATE}=+9V\$, but that will need another transistor to translate levels from 0V/+3.3V to +9V/0V (note the inversion):

simulate this circuit – Schematic created using CircuitLab
A low MCU output will switch the servo on, so I have tied R2 to +3.3V to ensure the servo remains off when the MCU output is at high impedance during power-on initialisation.
Usually this kind of high-side switching would be performed by a P-channel MOSFET, but that would still require level-translation. You are only able to continue to use the N-channel IRLZ44 because you have access to a potential significantly greater than +6V, which I have exploited above.
Diode D1 might not be necessary, but it's never a bad idea to protect the switching transistor from potentially inductive loads.
That's not all. Even if you manage to switch the servo perfectly on and off, so it sees the entire 6V supply when on, you are still supplying a PWM signal of only 0V/3.3V. Are you sure that this is sufficient amplitude for a servo powered from +6V?
This is how to use a P-channel MOSFET as a high-side switch:

simulate this circuit
Both transistor stages are inverting, so now a +3.3V MCU output will switch on the servo. I have moved R2 to reflect this change. You still require M2, to translate from 0V/+3.3V to +6V/0V. There's no need to use the +9V supply, since M1 is connected common-source, and is no longer a source-follower.