B737-700 captain reported an near midair collision while departing bur.

Narrative:

Departing bur on runway 15; we were assigned the heading of 210 degrees and an initial altitude clearance limit of 4000 ft. I believe it was just prior to 3000 ft that ATC asked us to expedite our climb to 4000 ft for traffic which was at 3000 ft. Shortly thereafter; with urgency and his voice; the controller told us to turn immediately right to the heading of 340. The first officer started to turn the aircraft while I was looking at TCAS. The intruder target was red and in close proximity to our aircraft. At this time; I thought it was necessary to intervene and increase the rate of turn and increase back pressure on the yoke to further increase our rate of climb; which I did. We were subsequently cleared to FL190 and on course; resuming our flight. Quite frankly; this event scared the hell out of me; and I would love to know how close we actually came. We never saw the intruder aircraft.

Original NASA ASRS Text

Title: B737-700 Captain reported an NMAC while departing BUR.

Narrative: Departing BUR on Runway 15; we were assigned the heading of 210 degrees and an initial altitude clearance limit of 4000 ft. I believe it was just prior to 3000 ft that ATC asked us to expedite our climb to 4000 ft for traffic which was at 3000 ft. Shortly thereafter; with urgency and his voice; the Controller told us to turn immediately right to the heading of 340. The First Officer started to turn the aircraft while I was looking at TCAS. The intruder target was red and in close proximity to our aircraft. At this time; I thought it was necessary to intervene and increase the rate of turn and increase back pressure on the yoke to further increase our rate of climb; which I did. We were subsequently cleared to FL190 and on course; resuming our flight. Quite frankly; this event scared the hell out of me; and I would love to know how close we actually came. We never saw the intruder aircraft.

Data retrieved from NASA's ASRS site and automatically converted to unabbreviated mixed upper/lowercase text. This report is for informational purposes with no guarantee of accuracy. See NASA's ASRS site for official report.

Attributes
ACN	1603255
Time
Date	201812
Local Time Of Day	1201-1800
Place
Locale Reference	BUR.Airport
State Reference	CA
Environment
Light	Daylight
Aircraft 1
Make Model Name	B737-700
Operating Under FAR Part	Part 121
Flight Phase	Climb
Flight Plan	IFR
Aircraft 2
Make Model Name	Any Unknown or Unlisted Aircraft Manufacturer
Person 1
Function	Captain Pilot Not Flying
Qualification	Flight Crew Air Transport Pilot (ATP)
Experience	Flight Crew Last 90 Days 451
Events
Anomaly	ATC Issue All Types Conflict NMAC