Is this pulse transformer in saturation?
Saturation looks likely BUT as Tom says - could be the transistor not having enough current sinking capability.

What happens if you change R3 to 1k?



How much inductance do you need for the transistor to have enough drive?:

At Vcc = 5V then I_R3 ~= (Vcc-Vbe)/R3 ~= (5-0.6)/10k ~~= 0.4 mA.
For V3 Beta = 100-200
I_V3 max = Beta x Ib = 40-80 mA.
Say 100 mA to be "kind".

I primary ~= Vp x t / Lp 
or Lp ~= Vp x t / Ip for core to just saturate in time t.
For 500 uS pulse, for core to just reach saturation
       Lp = Vp x t / Ip = 5 x 500 uS / 0.1 A = 25 mH
So you need a 25 mH effective primary inductance to remain within the transistor's drive capability.



1N4007 is not very suitable in this application.
1N514x  (x depends on voltage)or even 1N4148 would work better.