Waves of Western sanctions targeting Russian banks are hampering the country’s ability to conduct business, but efforts continue to be constrained by Europe’s reliance on Russian oil and gas.

The campaign to isolate Russia financially over the war in Ukraine has blacklisted most of Russia’s largest banks and taken aim at oligarchs. It also has targeted President Vladimir Putin’s daughters, with the U.K. on Friday saying it would join the U.S. and European Union by freezing any assets they have in the country and subjecting them to travel bans.

The EU on Friday also blacklisted four banks already banned from the Swift global payments system, including Russia’s second-largest bank, VTB, according to diplomats. VTB has already been sanctioned in the U.S. and the U.K.

But some big lenders and many small ones so far have been spared. Most Russian banks also continue to be connected to Swift, a financial-messaging infrastructure that links them to other banks around the world. That has left open a link to Asia and the Middle East, where Russia has mostly been free from sanctions, and has avoided a major disruption of energy supplies to Europe.

“It’s both extraordinary and groundbreaking the amount of sanctions imposed. But in many aspects, it is the tip of the iceberg,” said John Smith, a partner at law firm Morrison & Foerster in Washington, D.C., and former director in the U.S. Treasury’s Office of Foreign Assets Control, which enforces sanctions. The Ukrainian government is one of Mr. Smith’s clients.

In the latest round of sanctions this week, the U.S. and the U.K. ordered a full blockage of Sberbank, Russia’s largest bank, which holds about a third of all banking assets. They have also cut off Russia’s largest privately owned lender, Alfa Bank. Altogether, more than a dozen Russian banks have been blacklisted by at least one of the Western countries since the start of the war. The U.K. has fully sanctioned all the largest ones.

But holes remain. Several dozen Russian banks, although small, haven’t been sanctioned. The U.S. so far hasn’t targeted Gazprombank, a lender that has done business with close associates of Mr. Putin and has been embroiled in money-laundering troubles in Europe.

A woman leaves a Sberbank branch in Moscow.

Photo: natalia kolesnikova/Agence France-Presse/Getty Images

The EU also hasn’t touched the bank, and so far has held off on sanctioning Sberbank. European diplomats say the two lenders are a conduit to payments for Russian oil and gas, which member countries rely heavily on for energy consumption.

The EU imports more than $900 million worth of Russian oil and gas daily, according to Simone Tagliapietra,

a senior fellow at the Bruegel think tank in Brussels. Russian coal, which the EU has decided to stop buying, accounts for only $16 million in daily revenue for the country.

“Europe’s key weakness is a lack of geopolitical or risk perspective on energy policy, at national or at EU level,” said Karel Lannoo, chief executive of the Center for European Policy Studies, a Brussels think tank. “We are so bought into the system.”

The EU’s asset freeze on four Russian banks was part of a fifth round of sanctions the bloc has imposed on Russia since its invasion of Ukraine.

The latest measures, set to become effective later Friday, include a ban on Russian coal imports, a range of export bans and the addition of dozens of sanctions targeted at prominent individuals such as the sanctions on Mr. Putin’s daughters and leading Russian businessman Oleg Deripaska, officials said. The bloc also imposed a series of import bans including a ban on the import of Russian vodka and caviar.

A spokesman for the EU’s executive arm, the European Commission, said its sanctions are targeted at banks that most closely cooperate with the Kremlin.

As the commission worked on the latest round of sanctions earlier this week, a group of more than 200 lawmakers in the European Parliament called for all Russian banks to be disconnected from Swift, claiming anything less would be a weak response to the killing of civilians in Ukraine, particularly in Bucha.

“Progressive packages of sanctions with an autocrat doesn’t work,” said Guy Verhofstadt, a former Belgian prime minister who signed the letter. He said the latest round of sanctions by the EU were “ridiculous.”

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Sanctions experts and officials say the measures are having a real effect.

“Altogether, the comprehensive financial sanctions on Russia and Russian businesses and oligarchs are beginning to produce the intended effect of isolating Russia from the global financial system,” said Lisa Ledbetter, a financial industry partner at law firm

Reed Smith and a former attorney at the U.S. Treasury.

The country is struggling to import goods and materials, disrupting its supply chain and slowing down its economy. The European Bank for Reconstruction and Development last week said it expected Russia’s economy to contract 10% this year and suffer a prolonged period of stagnation.

More sanctions could be imposed as the war continues. Banning more banks from the Swift system, for instance, remains an option, according to EU diplomats. So far seven banks have been cut off, but VTB was the only large bank.

Ms. Ledbetter said that targeting the remaining big Russian banks should be carried out carefully and incrementally.

“Banning additional Russian banks [from Swift] could have collateral consequences for the global interbank, commodity and energy markets and could disrupt international financial integration,” she said.

As long as Europe and others remain dependent on Russian energy and other exports, Mr. Smith said sanctions on Russia are like delicate surgery. “You have to carefully use the scalpel without killing the patient, in this case, the international community,” he said.

—Max Colchester and Laurence Norman contributed to this article.

Write to Patricia Kowsmann at patricia.kowsmann@wsj.com