Air carrier Captain reported a potential near miss with another aircraft during climb. The Captain initiated a maneuver to avoid the other aircraft then resumed normal operations.

Narrative

The review team concluded a report that I had filled for a TCAS RA during a visual approach into BUR with Aircraft Y; that required evasive action the night of Day 0. I had another near collision event with multiple aircraft departing BUR in day VMC conditions. I was pilot flying (PF) and we had departed Runway 15 on the OROSZ2 Departure with LNAV/VNAV CMD A ON SQUAK of XXXX; we were honoring sterile cockpit procedures. I had the window open with 210 kts. in order to expedite our climb; which is always my plan to reduce the threat of general aviation traffic conflicts and gain altitude to receive a turn north with ATC. However; for some reason ATC was not issuing a turn clearance for some time. I remember thinking how unusual it was to have a clear view of LAX; as we were approximately 10 to 15 miles southwest of BUR; before ATC issued a turn to a 360 heading. We began the turn and were climbing through 6;000'; when ATC issued a Traffic Alert of a general aviation aircraft at 11 o clock at 6;500' and to expedite our turn." I ensured the aircraft was turning at the maximum 30-degree standard rate turn while we attempted to acquire the traffic visually. ATC then changed their mind and recommended we level off; which I promptly hit ALT HLD on the Mode Control Panel (MCP). However; due to our climb rate; the automation stopped our climb at 6;400'. ATC then issued another clearance to a fix. We continued our turn away from the traffic and additional clearance to a higher altitude was directed. To my surprise; ATC then issued another traffic conflict at our 12 o'clock position and above our altitude. I saw the target on TCAS. I believed Safety of Flight to be in question. Our closure rate; distance and altitude presented a significant collision hazard and I only had moments to take immediate action. I made the decision to not rely on last second ATC instructions or wait for a potential TCAS escape guidance and instead took immediate action to avoid a potential collision. I disengaged all automation and aggressively pitched the aircraft to 20-degrees of pitch and added near maximum thrust. As the nose of the aircraft pitched above the aircraft; I lost visual contact of the aircraft. No TCAS Alerts or RAs resulted and when it appeared we were clear of the aircraft I resumed normal operations and the rest of the flight was uneventful. No deviations from ATC clearances were apparent and ATC did not raise any concern.Preventative Measures: The First Officer (FO) and I debriefed the event later. We believed it unreasonable for ATC to direct many conflicting clearances and amendments to instructions to a transport category airline turbojet. We simply cannot maneuver to the extent ATC was expecting. We questioned why ATC had delayed our turn after departure and especially why ATC turned us towards two aircraft. It's our synopsis in this day and age with ADSB technology in general aviation aircraft that ATC knows the position of these aircraft and while their primary duty is to separate IFR from IFR traffic changes need to be made to improve safety in SoCal's procedures and or airspace parameters."