[RPG] How to a spell caster increase the probability that they can cast a spell from a scroll that is a higher level than they can cast

The simplest way is to ask for guidance or use inspiration

A druid or cleric in the party (or a character with guidance due to Magic Initiate Feat) can use the guidance cantrip to boost the DC check by 1d4. I will confine my answer to your simpler example, though you can see how it might work in more complicated situations.

From what I understand, even a level 1 Wizard can try to read a wish scroll. But how can an adventurer increase their likelihood of doing so?

The problem: you don't want to waste the spell on the scroll

If the spell is on your class’s spell list but of a higher level than you can normally cast, you must make an ability check using your spellcasting ability to determine whether you cast it successfully. The DC equals 10 + the spell’s level. On a failed check, the spell disappears from the scroll with no other effect. (SRD, p. 232, spell scroll, bolding mine)

Seek sources of advantage for the roll

If another character is available with the enhance ability spell (3rd level character or higher, 2d level spell) then casting that on the Wizard before the attempt will help.

Fox’s Cunning. The target has advantage on Intelligence checks.

A point of inspiration, if the Wizard has one, is another source of advantage.

If you have inspiration, you can expend it when you make an ... ability check. Spending your inspiration gives you advantage on that roll. (Basic Rules, p. 37)

If the Wizard has the lucky feat, that allows for an improved chance on the roll: in the simplest case, advantage by rolling a second d20

Whenever you make an attack roll, an ability check, or a saving throw, you can spend one luck point to roll an additional d20.

As was shown in this answer, advantage improves the chances by 3 or more points, added to that 1d4: guidance plus advantage. Getting Bardic Inspiration rather than guidance to help would add 1d6 at low levels. Why not try and get both?

Bardic Inspiration (so that's why we have a bard in the party ...)

If the party has a bard in it, adding bardic inspiration would boost the 1d4 roll by another 1d6. (At lower levels. As levels go up, the benefit increases: d8 at 5th; d10 at 10th; d12; at 15th and higher). Granted, at that point the Wizard in question is going to have a higher Int bonus already.

As you can see, the answer for a given situation and a given spell DC will depend on the party's make up, in terms of how many other bonuses one can add to the d20 roll.

The surest way is with Divination School's Portent (if available)

If the wizard takes the Divination School at second level, the class feature Portent may be a huge boon here, but it depends on what die rolls are being held for the occasion. (see below)

Example problem: 4 out of 5 times guidance + bardic inspiration succeeds.

To illustrate your opening example: what are the the chances to achieve a target number of 19 (wish, 9th level spell) using help from this answer and assuming an Intelligence score of 16 (+3 bonus) for our 1st level wizard.

1. Unmodified: the wizard has a 25% chance to succeed on that DC 19 ability check. (Needs to roll a 16 or higher).

2. With advantage, 1st level wizard has to roll a 16 or better: 43.75% chance.

3. With guidance, an increase of 1-4, the range needed to roll is from 15-12: 51% to 69.75% chance to succeed.

4. Add in a 1d6 of bardic inspiration to guidance, and the odds get very good. Those two dice will add between 2 and 10 to the roll. With advantage between 57.75% chance to 93.75% chance (Roll a 6 (best case) through a 14 (worst case) to succeed).

   The first level Wizard can expect to succeed roughly 4 times out of 5 when combining guidance and bardic inspiration with advantage. The average of 1d4 is 2.5; the average of 1d6 is 3.5, for a rough combined "average" of 6 for simplicity's sake; the Wizard needs to roll a 10 or better, with advantage: 79.75%. Close enough to 4/5 for these purposes. Or, 78.71% using anydice for the whole best-of-2d20 + d4 + d6 + 3 roll (thanks @KamilDrakari)

For your mid-level case: it depends

a mid-tier conjuration wizard attempting to cast some sort of high level spell via spell scroll, although the school of magic they follow does not have to be conjuration. Class features, spells and magic items are of most interest, although the wizard does not have to be by himself.

You can use the same process I outlined above, modified by whatever magic items, spells, and class features that you have at hand to boost the result of that roll. Whatever increases your generic INT Ability check roll results, or boosts your INT bonus (INT 20 = +5 bonus), will increase your chances of success.

If the spell on the scroll is on the Bard spell list, or your scenario unique spell is one a Bard is eligible to cast, Jack-of-All-Trades helps the Bard to read the spell from the high level spell since it is an ability check. At 9th level, with proficiency bonus of +4, that's +2 that can't be gotten in another way.

Starting at 2nd level, you can add half your proficiency bonus, rounded down, to any ability check you make that doesn’t already include your proficiency bonus. (emphasis mine. Thanks to @E.Downer for the tip; even though your unique case specifies mid-level conjuration Wizard).

Portent Deserves Special Mention

Divination school's Portent class feature is a huge benefit since you know the result ahead of time. If you already know the die roll, you can adjust up from that result, if needed.

When you finish a Long Rest, roll two d20s and record the numbers rolled. You can replace any Attack roll, saving throw, or ability check made by you or a creature that you can see with one of these foretelling rolls. You must choose to do so before the roll, and you can replace a roll in this way only once per turn. (PHB, Divination, level 2 class feature).

Let's say that your mid level wizard has a 20 Int at level 8. Portent rolls result in 4 and 11. The Wizard adds 5 to any roll due to Intelligence bonus. Pick the 11: only 3 more points are needed to achieve that 19. Lucky, advantage, inspiration, bardic inspiration, guidance ... get them from somewhere. The sources are varied with group composition and items at hand. Any item that increases the Intelligence modifier or allows a boost to a roll will increase the chances of success.

It's all or nothing! We get one shot at this! ("Superadvantage"-Lucky Feat)

You'll want the Lucky feat, and another source of advantage (the enhance ability spell or an Inspiration point grant that) to get Lucky's super advantage. That's where you get to roll 3d20, not 2d20, and add all of the bonuses from your other sources to the best roll.

The curious case of the disadvantage and Lucky

If the character has the Lucky feat, there is a way to turn disadvantage into super advantage with that feat. Details here. For example, the wizard could find a way to accrue a level of exhaustion, which gives disadvantage on ability checks, and making the attempt with lucky grants super advantage as above. This is an odd quirk in the lucky feat.

LUCKY

For example, if you have disadvantage on your attack roll, you could spend a luck point, roll a third d20, and then decide which of the three dice to use. You still have disadvantage, since the feat doesn’t say it gets rid of it, but you do get to pick the die.

There has since been a tweet from Jeremy Crawford that he's changed his mind, but that text is still up at the WoTC web site, the errata to the PHB shows no change to the feat, and the published Sage ADvice Compendium, 2017, retains the above text. Before trying this, work with your DM; if you are DM discuss this with your players ahead of time if one of them has the Lucky feat. Arrive at a ruling that makes sense for your table. (Thanks to @AntiDrondert for the reminder).