Google is indeed your friend, and the fact that you are from US, an English-speaking country, is a great plus.

Now let me say. When visiting other countries, you must check with your doctor and local regulations because not all drugs can be sold by a random stranger walking into pharmacy. You may need an internationalized prescription, which is handled by the pharmacist

The key is finding the generic name of the drug, which often matches the active ingredient(s)' name, with the help either of the package or online. Secondarily, you will have to deal with the dose.

Suppose you have a prescription for... random... Solurex.

1. Find the active ingredient, in English

A quick search for [pharmaceutical name] generic name reveals that Solurex is based on something called Dexamethasone.

You just achieved 98% of your job!!. Now you just have to pick the correct local drug.

2. Translate the name into any language

With the help of an online translator?

• Spanish: Dexamethasona
• Norwegian: Deskametason
• Russian: ????????????
• Hebrew: ????????
• Simplified Chinese: ????

Of course, if your pharmacist speaks English they can understand what medication to serve you. Remember that the pharmacist wants to know both the active ingredient(s) and its dose. I helped on finding the ingredient name.

Most if not all pharmaceutical products are equivalent to their generic drug despite a different commercial name. So you can buy just any Dexamethasone anywhere even if it's sold under different names, as soon as the dose matches your prescription.

Bottom line

Don't expect any drug to be available easily. Jurisdictions vary, certain drugs couldn't be authorized at all in the target market. Simple everyday products like Aspirin (which is the commercial name), paracethamol (which is the generic name that you can find e.g. as Beechams Flu Plus Hot Lemon) are of course widely available and mostly require no prescriptions. For more specific medications, e.g. anti-cancer, anti-diabetes, life-saver, you may have to buy home and carry with you. Indeed, this is the safest way.

As a last resort you could always try finding a local doctor to give you a prescription but that could prove difficult or expensive depending on your destination.

I found the following resource to find the name of the equivalents of a US drug in other countries: https://www.drugs.com/international/

The Drugs.com International Drug Name Database contains information about medications found in 185 countries around the world. The database contains more than 40,000 medication names marketed outside the USA and is presented in multiple languages.

Example: https://www.drugs.com/international/acemetacin.html

Another useful resource: https://pillintrip.com/ (not restricted to pills despite its name, e.g. it also contains creams)

The existing answer is good, but I'd like to clarify the different types of drug name.

Each drug has a generic name, and often one or more brand names under which it's sold.  Sometimes it's sold under the generic name directly (e.g. by smaller pharmacies, supermarket own brands, online, etc.).

Commonly-known and -available brands vary across the world.  Generic names, though, are much more consistent; they even look and/or sound similar across different languages, specifically to help in cases like this.  (They also uniquely identify a drug, unlike brands which can contain multiple active ingredients.)

So the important thing is to find the generic name.  (If you have a product, it'll be in the list of active ingredients.)  Pharmacists and other medical professionals know generic names, so you can ask for it directly.  And you can look for it on other products.

The painkiller under discussion is a very unusual case, as it has two different generic names in English, as well as many different brand names.  Its International Non-proprietary Name (INN) is ‘paracetamol’, but in the USA and a few other countries it's known as ‘acetaminophen’.  Some examples:

I don't know about other countries, but in the UK the generic name ‘paracetamol’ is the one it's most commonly known by, so that's all you'd need to know.

(In general, I get the strong impression that people in the USA prefer using brand names, even when not referring specifically to products bearing that brand; while in the UK we tend to avoid them in those cases, so as not to be overspecific — and not give free advertising!  But of course that's a discussion for another site.)

A search on Wikipedia will often yield exactly the information you seek.

You can search for the chemical name or trade name. Wikipedia is smart enough to figure out what you mean, and will typically take you to the correct page.

On the page it presents, trade names will often be presented for multiple geographic regions.

Here is a good example: A search on Wikipedia for acetaminophen or paracetamol will both lead you to this Wikipedia page:
https://en.wikipedia.org/wiki/Paracetamol

On that page, you can see that Tylenol and Panadol are both common trade names. That page even links you to another page with more details:
https://en.wikipedia.org/wiki/Paracetamol_brand_names

On that page you will learn about over 50 different trade names for that chemical, as well as different formulations, delivery mechanisms, and packaging. It also lists medications that contain the primary chemical combined with other chemicals.

One advantage of Wikipedia is that it is often more up-to-date than other sites (but not always.. finding multiple sources of data is always essential) and another is that it contains content from around our planet.

Others have commented on how to find equivalent drug names, I have another very important tidbit to add that is related to the problem.

THE LEGALITY OF YOUR DRUGS IN YOUR DESTINATION

Many countries have comprehensive lists of drugs (subject to immediate change without notice,) that are illegal to possess or bring into the country!

For example Japan (from: https://jp.usembassy.gov/services/importing-medication/)

Many common medications and over-the-counter drugs in the United States are illegal in Japan. It does not matter if you have a valid U.S. prescription for a medicine/drug which is illegal in Japan: if you bring it with you, you risk arrest and detention by the Japanese authorities. The U.S. Embassy and Consulates in Japan do not maintain a comprehensive list of prohibited medications or ingredients.

Comprehensive information is available only from the Japanese government and is subject to change without notice. All travellers are encouraged to check before travelling to Japan at Japan’s Ministry of Health, Labour, and Welfare (MHLW) website, including reviewing FAQ, or to email yakkan@mhlw.go.jp.