No, dispel magic will not remove the turrets

Arcane Turret says:

At 3rd level you learn how to create a magical turret. With your smith's tools in hand you can take an action to magically summon a Medium turret [...]

So the turrets are magical and are created by taking a specially defined action as part of a class feature. However they are not spells or created by a spell effect.¹

This is important because dispel magic only has any effect on things that are spells:

Choose one creature, object, or magical effect within range. Any spell of 3rd level or lower on the target ends.

Since magical turret is not a spell, dispel magic won't do anything to them (unless they happen to have an unrelated spell on them).

This aligns with guidance from the Sage Advice Compendium:

Can you use dispel magic to dispel a magical effect like a vampire’s Charm ability or a druid’s Wild Shape?

Dispel magic has a particular purpose: to break other spells. It has no effect on a vampire’s Charm ability or any other magical effect that isn’t a spell. It also does nothing to the properties of a magic item. (SAC v2.3 p 14)

^{1 - You can see this Q&A for an in-depth discussion for what counts as a spell. In short: nowhere does the feature say anything about a spell, or say you "cast" anything, or give this ability a spell name or spell block listed anywhere. Without those things, this can't be a spell.}

The second scenario is correct: Spell slots = resources

Once you create a cannon {using an action+tools}, you can't do so again until you finish a long rest or until you expend a spell slot of 1st level or higher

The key to this is the "again", the "until" and the "expend" - there is a resource cost for summoning a subsequent cannon after the free "first cannon of the day." The artificer either waits for a long rest to get another free cannon, or, the artificer expends a spell slot to get another cannon before that long rest recharges the free cannon. The creation of a cannon does not take two actions; - the Artificer spends a spell slot to create the next cannon(s). (If after "or higher" they had added "to create one" it would have better matched how they phrased it in the UA - Jeremy Crawford's tweet indicates that they intended it to work the same way as in UA).

The subsequent cannons resemble the Ranger "Primeval Awareness" class feature ability in that the character expends a spell slot to activate a class feature.

Primeval Awareness
Beginning at 3rd level, you can use your action and expend one Ranger spell slot to focus your awareness on the region around you. For 1 minute per level of the spell slot you expend, you can {snip}...

_{Sadly for the Ranger, they don't get the first use for free as the Artificer does. We now initiate a mass protest against WoTC for not loving the Ranger class}

A spell slot is used to create the next* cannon(s).

Cannons last for an hour or until the cannon reaches 0 HP.

It disappears if it is reduced to 0 hit points or after 1 hour. You can dismiss it early as an action. (RftLW, p. 59)

The order of operations would play out like this:

1. Create cannon (first cannon of the day). (Uses an action, no spell slot consumed)
2. Other things happen. (Likely combat, or time expires ... )
3. Same adventure day, you need another cannon, or, you want to change the kind of cannon you have (there are three kinds).
4. Expend a spell slot to create the next one

You could run yourself out of spell slots if you go cannon-happy

A likely reason for subsequent cannons costing resources: prevent the creation of multiple cannons that cost no resources. Compare case 2 with case 1, in terms of resources:

• Case 1: 1 cannon + as many cannons as spell slots + all those spell slots cast as spells.

• Case 2: 1 cannon + as many cannons as spell slots versus

  Case 1 is a significant resource boost for one adventure day for 6th level Artificer, for example

Designer Commentary/Clarification from Jeremy Crawford

Artillerist artificer's Eldritch Cannon feature says, "Once you create a cannon, you can’t do so again until you finish a long rest or until you expend a spell slot of 1st level or higher." In response to a question about whether spending a spell slot on anything else is sufficient to create a new cannon, Crawford replied: "Eldritch Cannon does let you spend a spell slot to create the cannon. That slot must be spent specifically on the cannon, not on something else." (From this tweet)

Experiential notes:

1. That's how we did it using the Artillerist from the final Artificer UA. This feature is roughly unchanged from that version of Artillerist.

2. IMO, this can be very powerful, but if the whole campaign is in Eberron, with its elevated power levels, it probably won't be that big of a deal.

3. The text from the last Artificer UA; this points toward some design intent.

   Arcane Turret
   {description} It disappears if it is reduced to 0 hit points or after 10 minutes. You can dismiss it early as an action. When you summon the turret, you decide which type it is, choosing from the options on the Arcane Turrets table. {more actions} . You can summon a turret once for free and must finish a long rest before doing so again. You can also summon the turret by expending a spell slot of 1st level or higher.

¹ A Protector cannon can be a lot better than cure wounds ...

You mentioned cure wounds as the spell, so I'll point out that the Protector cannon is often a better use of a level 1 spell slot than cure wounds.

Fun Protector Cheese: Set up a Protector with an action, and keep loading up your allies with 1d8+Int THP each round as a bonus action. Even if the Artificer burned a spell slot to do this, that's potentially a lot more HP than a "cure wounds" (1d8+Mod) or a healing word (1d4 +mod) cast once. With three allies, you can offer each of them 1d8+mod of damage prevention (THP) each round for the cost of a bonus action. And, if you have two fights within one hour, you keep covering them in THP to mitigate damage taken - but you didn't have to use any more spell slots. Note that the THP also applies to the cannon, which will reduce the chances that it gets reduced to 0 HP during the fight. (Until you get to higher levels)