Have you had the new battery load tested? It could be a bad battery even though new. Second is to verify all connections from ignition to battery to starter to alternator are all clean at terminals and tight. Also, check your grounds. Check the output from the alternator to verify the amperage and voltage are good. The alternator could have bad diodes or some other issue and not charging the battery properly and/or draining the battery when the engine is off.
Next, your starter switch appears to be connecting to turn on dash lights but the connection point for starting could be glitchy. You could take it off and repeatedly test it for continuity by pressing it, no or intermittent readings could be a bad switch. And your starter could be going and not always fully engaging when power gets to it. You could test the current to it when the switch is pressed and it doesn't start.
One of the 4 major parts; wires, battery, alternator, starter ( and switch), one is possibly your problem. Going to take some investigating to figure it out. Good luck.