The issue you are experiencing is often referred to as "cam out" if you should wish to research the topic any further (at least aside from that hands-on research experience you have already acquired).
Both Phillips and traditional straight bladed screw heads are very prone to cam out just by way of their design (in the case of Phillips it was intentional to prevent overtorquing the screw or bolt), hence a big part of the reason that alternatives like Torx, Pozi-drive, Robertson (square), hex/Allen head, etc., have all been created and marketed over the years for higher torque applications. Also be aware that these days many screws that look like Phillips are actually JIS (Japanese) standard (and increasingly Pozi-drive as well, but Pozi-drive can be easily recognized by the much smaller wings in between the four major wings forming a nearly square center opening) and using a normal Phillips bit in these screws will result in one cramming out very easily, chewing up the face of the screw. You can recognize JIS screwdrivers by a flat (in the smaller sizes) at the tip where a Phillips comes to a much sharper point by comparison. But the JIS/Phillips confusion results in a lot of headaches because using Philips screwdrivers in JIS screws leads to a rather poor fit prone to cam out when you try to apply much torque due to the deeper point of the Phillips driver, and because the two look so similar superficially we tend to mix up both the screws and their drivers in our parts and toolboxes without realizing such. You can generally count on getting away with using a JIS screwdriver in place of a Phillips when mating to a Phillips screw, but not the other way around since a Phillips driver can not fully mate with a JIS screw.
However, when you need to apply enough torque to risk cam out you also have another option besides the "lowly" screwdriver and that is using an electric (usually cordless) impact driver. Impact drivers can deliver a great deal more torque without cramming out because they utilize a high impulse hammering action that exploits the inertia provided by the tool's own weight and only rotates a short distance on each strike (further reducing the risk of cam out since the but can reseat between strikes). Additionally, they avoid the usual torquing against the user that one experiences when using a typical drill/driver or cordless screwdriver -- so much so that they can be comfortably used one handed even in high torque applications.
Hope this also helps to shed some light on the topic as I know it can be quite frustrating.
