Curling leaves are a common issue with lilacs, and such problems are likely to persist even with utmost care. There are certain plant diseases and bacterial infections that cause this condition. At times, you may need to adjust your watering schedule. Certain factors, like temperature changes, can also aggravate this issue.

Identifying the root cause of this phenomenon is vital in preventing further damage. The reasons could be due to improper care, environmental changes, or pests.

For your specific situation, the plant appears to have some type of blight, which is a bacterial infection.

Bacterial Infection
Bacterial infection or blight causes lilac leaves to curl, sometimes with brown, water-soaked spots or dry edges. It also manifests as brown/black lesions that can spread to the stem.

Pseudomonas species that commonly infect trees and other woody plants often cause bacterial blight in lilacs. Worse, they spread through the air or using infected apparatus on your plant.

https://extension.oregonstate.edu/news/how-recognize-treat-avoid-lilac-bacterial-blight
Attached at the bottom of this response are more articles which may be a help about this issue.

Beyond this, it is possible that there are other factors involved with what you are seeing with your plant. These are insect pests, fungal diseases, and some issues with environmental care such as pruning and fertilization.

Insect Infestation
Certain pests, especially the leaf miner, cause curled leaves in lilacs. They’re a type of fly about 1-2 mm in size, but their larval stage causes damage to lilac leaves.

This is because leaf miner larvae penetrate the leaves, burrowing or mining through the plant tissues. As a result, the leaves appear with serpentine-like or irregularly shaped white patches.

With the damage progressing, the leaves desiccate, curl up, and eventually fall off.

Aside from making the leaves look distasteful, heavy infestation impacts the plant’s overall health, too. Once the leaves are heavily damaged, chlorophyll declines, so the plant can no longer generate enough energy to sustain itself. It leads to poor plant health and, eventually, death.

https://hortsense.cahnrs.wsu.edu/fact-sheet/lilac-lilac-leafminer/

The Lilac borer is the most common lilac aggressor and is typically found on older branches first. Adult borers are wasp-like moths with clear wings and red heads, while larvae are white-pink caterpillars with red-brown heads.

Look for cracked bark, broken branches, and/or stem holes that may exude sap or a sawdust-like excrement to indicate lilac borer infestation.

If borers are caught early, they can be effectively managed by pruning off limbs that are being attacked, so careful monitoring is key to addressing this pest. Just know that recovering lilacs may need extra watering and attention for a while.

https://www.canr.msu.edu/news/lilac-borer-is-a-pest-that-may-weaken-shrubs-over-time

Fungal Diseases
When you see white, powdery spots or patches on your plant’s leaves or stems, powdery mildew could be the culprit. At the same time, heavy infection will eventually lead to leaves curling.

It’s because powdery mildew is a damaging fungus that the plant can acquire through infection from another plant.

This fungi is easily transmissible and is airborne, so it’s likely that overcrowding your garden or nursery can transfer the fungi from one infected plant to another. Another factor that encourages powdery mildew growth is increased humidity.

This is due to the fact that too much moisture (over 90% humidity) supplements the growth of fungal spores.

Temperatures within 60 to 86° and limited sunlight exposure or too much shade encourage spore formation, too.

https://extension.unh.edu/blog/2018/09/what-should-i-do-about-powdery-mildew-my-lilacs

Too Much Heat and Sunlight
Extreme temperatures and prolonged sun exposure dry up lilac leaves and cause them to curl. Note that lilacs still develop curled leaves even with substantial watering but with excessive heat or sunlight, so they should receive a balance of both elements.

Improper Nutrition
Your lilac leaves curling may also be due to insufficient minerals such as nitrogen. This element is essential to maintaining plant health and leaves as well as encouraging flower growth. Further, fertilizing in early spring is most favorable.

How Do You Treat Curling Lilac Leaves?

Here are various ways to treat curled leaves in lilacs depending on which factors are causing the problem:

1. Treat Plant With Pesticide or Horticultural Oil
To treat pest-infested lilacs, use pesticides that contain pyrethroids because they work best in killing leaf miners.

It’s also essential to treat the leaves in the spring with dormant horticultural oils to both kill any eggs that may have remained from the previous season and to prevent further infestations from occurring. It’s also a natural and less toxic alternative to commercial pesticides.

2. Prune the Lilac Bush
Pruning Lilac Bushes Can Help Prevent Leaf Curling.
Pruning your lilac bush is an excellent choice, especially if signs of other diseases are also present, such as wilted or droopy leaves with either black or brown spots.

To prune lilacs, look for severely damaged branches and remove them using a clean or disinfected pair of pruning shears/scissors.

At the same time, properly dispose of branches, flowers, and leaves that have been pruned out to discourage the spread of diseases to other healthy and thriving plants.

3. Use Fungicides
Fungicides are effective in treating powdery mildew and further spread of disease by killing the fungal spores.

In fact, combining pruning dead branches with a suitable fungicide is an excellent strategy to stop the spread of the disease while encouraging new growth.

4. Plant in a Sunny Area and Avoid Overcrowding
Replanting the lilac bush in a sunnier or warmer area helps, although this technique can be difficult if the bush is too large. These plants need at least six hours of sunlight daily for optimal growth.

Remember to leave some space between each plant (about three feet apart) for good airflow, too.

5. Avoid Overwatering or Underwatering
Keep the plant’s soil moist but not mushy to prevent drought stress due to underwatering or powdery mildew because of too much moisture. To do this, wait until the upper layer (1-2 inches) of the soil is dry before watering again.

At the same time, you should pour water directly into the soil instead of getting the foliage or flowers wet, which also encourages bacterial or fungal growth.

6. Use a Well-Balanced Fertilizer
Supplement your plant’s health by incorporating a 10-10-10 fertilizer before their growth season (early spring). Note that you should avoid high-nitrogen fertilizers as they can impair the plant’s flowering capacity and cause excessive leaf growth.

These additional articles may of help:
https://extension.umd.edu/resource/lilac-identify-and-manage-problems/agnr.umd.edu/resource/lilac-identify-and-manage-problems
https://extension.psu.edu/lilac-diseases
https://extension.illinois.edu/blogs/good-growing/2021-08-05-whats-wrong-my-lilac-summer-foliar-diseases-lilac