This is not accurate. Read the spell description - the creature thinks and behaves as if the phenomenon is real. The example of a victim falling through a Phantasmal bridge happens SPECIFICALLY because the victim acts appropriately on their belief. A dragon (or anything else) caught in a Phantasmal Force-Web feels the stickiness of the webbing on its body, feels the tension of the strands on its limbs, smells the adhesive (or whatever spider web might smell of) coating the webbing, rationalizes ways in which it is entangled. It doesn't just try to flap its apparently entangled wings out of the webbing any more than a PC typically walks straight into a wall of force, ice, or stone that appears when a caster waves their hands without good reason to suspect an illusion. Escape attempts from nets, grappling foes, restraining spells use an action, not mere movement. A strong creature might try to ram an apparent wall with the intention of knocking it down, might ignore an elemental that doesn't seem to inflict much damage, might examine a creature after an attack or two fails to hurt it, might investigate webbing that appears to be holding it without being attached to any floors or ceilings, or why a cage is floating in mid-air.
Not a flying dragon.
Maybe one standing on the ground, but even then, how does the DM adjudicate it?
Most dragons do not hover. The spell indicates inside it that there is a 10 foot cube and later on indicates that an illusion of a creature can only harm the target if the target is within the area of the spell. It's a non-mobile spell. No flying elementals. No putting a mobile cloud of darkness around the head of a dragon. In fact, the concept of a visual phenomena could be read to mean not spells, just water, lava, poison gas, etc. How does one adjudicate other spells within a Phantasmal Force spell? How do creatures make saves against a spell that does not exist? The spell does not state that it can emulate the power of other spells.
The dragon believes that the web is real, but gravity still works in the web example, just like it does on the bridge example. The dragon falls through the web, thinks that it avoided the web (just like a creature making its save thinks that it avoided a web against a real web spell), and continues on with a DM who rules that the Dragon stops trying to fly.
For a DM that rules that the Dragon keeps flying on, it flies through the web, again, thinking that it avoided it. The DM could state that the dragon slowed up (difficult terrain) but continued on.
There isn't actually anything there to hold the Dragon up in the air. There isn't actually a Dex saving throw for the Dragon to fail such that it thinks that it is stuck.
Put a 10x10x10 net around the flying dragon (course, dragons with lairs are huge), it still falls / flies through it. Nothing real is there. A huge or larger dragon on the ground would think that the net is real and might use its intelligence check action to try to break through it (not a strength check action) because the target knows what a net is and might think it is trapped (a huge dragon also might care less about a 10x10x10 net and ignore it, thinking that it is not enough of a hindrance, when the dragon moves on, the "net" falls off). But not a flying dragon that cannot hover. It is forced to move through the illusion and when it does (by accident), it thinks that it avoided the net in some way.
I don't personally think that a second level illusion spell should be able to emulate the power of other spells. Phenomenon should mean things like water or lava, not spells. It's too difficult to adjudicate spells because there is no actual web there and the target might be unaware of what the web spell does anyway, no actual faerie fire there (how does one adjudicate advantage by attackers in the mind of the target?), no actual darkness, etc. How does the target know how the fake spell works? How does the target emulate the effects of a spell? The Phantasmal Force spell is just a simple visual illusion in an immobile 10 foot cube that is easily overcome by falling through it, not an elaborate high level illusion.
Obviously, a given DM can adjudicate any way he wants to. But based on the examples and text within the spell, people here are trying to make that spell a lot more powerful than the designer's appear to have intended.