Are you tensing your arm before you punch, you should only contract your fist when you punch makes contact, I know that helped me.

Punches, should be thrown relaxed, it allows you more speed and makes you hit with more power.
Try tensing you muscles as you throw punches, then try relaxing your muscles and throwing combinations, notice a difference.
Keep them relaxed.

You will develop strength and endurance the more you practice, soon you will not notice your lead arm getting tired.
If it is a new activity it will take time for your body to adapt.

You should not be dropping your arm at any point during a bout or sparring, concentrate on your defense and conditioning.

Remember to include rest days in your training as it will allow faster recovery and give bigger improvements in your boxing skills and fitness.