'When you cast a spell' could mean 'at the moment you cast it', but probably does not
Most of the answers here arguing that Careful Spell cannot protect against Web explicitly interpret the phrase 'when you cast a spell' to mean on the turn you cast it.
Among those answers arguing that Careful Spell can protect against Web, Brad Szonye's points out that "In plain English, when can mean either the timing or circumstance of an event" and Scott makes the same point.
So, given that 'when you cast' could be interpreted two different ways, what context clues do we have for how it should be interpreted?
For that, I look to another metamagic feature, Heightened Spell, which says (emphasis mine):
When you Cast a Spell that forces a creature to make a saving throw to resist its Effects, you can spend 3 sorcery points to give one target of the spell disadvantage on its first saving throw made against the spell.
Why specifically call out its first saving throw, rather than simply saying "you can spend 3 sorcery points to give one target of the spell disadvantage on its saving throw"? If the "when you cast" clause meant that only saving throws made on the turn the spell was cast counted, then the only reason to indicate its first saving throw would be for spells on which more than one save is made by a target on the turn the spell is cast.
On the sorcerer spell list I am counting just one⚔ spell that calls for multiple saves on the turn it is cast, Earthquake, and then only if a target is also concentrating.
To me, it strains credulity to believe that the 'first saving throw' language was deliberately added to the Heightened Spell description so as to cover the effects of just one spell a sorcerer can first get at 15th level.
On the other hand, it seems very reasonable to me that the 'first saving throw' is meant to apply to sorcerer spells that provoke saves over multiples turns or on subsequent turns, such as⚔ Blindness/Deafness, Gust of Wind, Hold Person, and Web all starting at 3rd level, and later on Fear, Sleet Storm, Slow, Stinking Cloud, Confusion, Cloudkill, Hold Monster, Insect Plague, Delayed Blast Fireball, Incendiary Cloud, and Power Word Stun.
If you accept that the "when you cast" language in Heightened Spell indicates saves made on turns after the turn in which the spell is cast, it is not a long walk to accepting that the same phrase used for Careful Spell can also be applied to subsequent turns. If not, then you need to explain why the phrase has different meanings for the different metamagic abilities.
⚔I am currently working from the SRD and realize that there may be other spells in both categories. However, I doubt there are many more that provoke two saves for the same creature on the same turn the spell is cast.