What is a stronger alternative to "avoid"?

I think the simplest way to emphasize avoidance would be to use the word shun.

shun v. tr.
to keep away from; take pains to avoid.
_{See TFD Online}

Note the "take pains" in the definition. It suggests a strenuous avoidance, which should be what you're looking for.

Nota bene: To all those who subscribe to the narrow viewpoint that shun is archaic, or only ever used for people, or subject to other strictures, here are a few current links. Note that the first link is from the Washington Post and is no more than a few months old.

Will the new women in Congress embrace bipartisanship—or shun it?
Do you shun the use of autoclickers?
Dividend ETFs Tend to Shun Tech Sector—Barron's
CR Boldface: Shunning the use of titles
NM debtors tend to shun filing Chapter 13
Why did nomadic peoples shun the use of pottery?
... and my current favorite:
Consumers shun macaroni products (!)

Shun is not only commonplace but is frequently used in a variety of contexts by intelligent English speakers—especially when they want to make a strong statement about avoidance. This has been true for centuries and is still true today.

How about plain old Do not? It's not sexy, but it gets the point across unambiguously.

I'm jumping in with an edit...

Avoid setting the cat/house on fire.

Shun setting the cat/house on fire.

Abstain from setting the cat/house on fire.

Do not set the cat/house on fire.

Not sure how strong you want to be here, but, eschew is pretty strong.

From MW eschew:

eschew v.

to avoid habitually on moral or practical grounds

The expression to steer clear of something or someone sounds stronger, I think. It's oftentimes used in situations where you're advised to avoid something that can be very dangerous for you. Somebody advising you against doing drugs would be one good example. The following is how the Cambridge Dictionary defines this expression:

to avoid someone or something that seems unpleasant, dangerous, or likely to cause problems

Example sentence:

They warned their children to steer clear of drugs.