diff --git a/packages/playwright-core/browsers.json b/packages/playwright-core/browsers.json
index d0b427f41b8ae..dd9fd6058878c 100644
--- a/packages/playwright-core/browsers.json
+++ b/packages/playwright-core/browsers.json
@@ -27,7 +27,7 @@
     },
     {
       "name": "webkit",
-      "revision": "2061",
+      "revision": "2062",
       "installByDefault": true,
       "revisionOverrides": {
         "mac10.14": "1446",
diff --git a/tests/library/browsercontext-viewport-mobile.spec.ts b/tests/library/browsercontext-viewport-mobile.spec.ts
index 8160d5601e314..1f897f60fa9b9 100644
--- a/tests/library/browsercontext-viewport-mobile.spec.ts
+++ b/tests/library/browsercontext-viewport-mobile.spec.ts
@@ -15,6 +15,7 @@
  * limitations under the License.
  */
 
+import os from 'os';
 import { browserTest as it, expect } from '../config/browserTest';
 
 it.describe('mobile viewport', () => {
@@ -54,7 +55,8 @@ it.describe('mobile viewport', () => {
     }
   });
 
-  it('should be detectable by Modernizr', async ({ playwright, browser, server }) => {
+  it('should be detectable by Modernizr', async ({ playwright, browser, server, browserName, platform }) => {
+    it.skip(browserName === 'webkit' && platform === 'darwin' && parseInt(os.release(), 10) === 22, 'detect-touch.html uses Modernizr which uses WebGL. WebGL is not available in macOS-13 - https://bugs.webkit.org/show_bug.cgi?id=278277');
     const iPhone = playwright.devices['iPhone 6'];
     const context = await browser.newContext({ ...iPhone });
     const page = await context.newPage();
@@ -63,7 +65,8 @@ it.describe('mobile viewport', () => {
     await context.close();
   });
 
-  it('should detect touch when applying viewport with touches', async ({ browser, server }) => {
+  it('should detect touch when applying viewport with touches', async ({ browser, server, browserName, platform }) => {
+    it.skip(browserName === 'webkit' && platform === 'darwin' && parseInt(os.release(), 10) === 22, 'Modernizr uses WebGL. WebGL is not available in macOS-13 - https://bugs.webkit.org/show_bug.cgi?id=278277');
     const context = await browser.newContext({ viewport: { width: 800, height: 600 }, hasTouch: true });
     const page = await context.newPage();
     await page.goto(server.EMPTY_PAGE);
diff --git a/tests/library/browsercontext-viewport.spec.ts b/tests/library/browsercontext-viewport.spec.ts
index 360cc5815df9d..4ba1e2f52d448 100644
--- a/tests/library/browsercontext-viewport.spec.ts
+++ b/tests/library/browsercontext-viewport.spec.ts
@@ -93,7 +93,8 @@ it('should emulate availWidth and availHeight', async ({ page }) => {
   expect(await page.evaluate(() => window.screen.availHeight)).toBe(600);
 });
 
-it('should not have touch by default', async ({ page, server }) => {
+it('should not have touch by default', async ({ page, server, browserName, platform }) => {
+  it.skip(browserName === 'webkit' && platform === 'darwin' && parseInt(os.release(), 10) === 22, 'detect-touch.html uses Modernizr which uses WebGL. WebGL is not available in macOS-13 - https://bugs.webkit.org/show_bug.cgi?id=278277');
   await page.goto(server.PREFIX + '/mobile.html');
   expect(await page.evaluate(() => 'ontouchstart' in window)).toBe(false);
   await page.goto(server.PREFIX + '/detect-touch.html');
diff --git a/tests/library/modernizr.spec.ts b/tests/library/modernizr.spec.ts
index 5f0e165c65922..327a3c4169556 100644
--- a/tests/library/modernizr.spec.ts
+++ b/tests/library/modernizr.spec.ts
@@ -32,7 +32,8 @@ async function checkFeatures(name: string, context: any, server: any) {
 
 it('safari-14-1', async ({ browser, browserName, platform, server, headless, isMac }) => {
   it.skip(browserName !== 'webkit');
-  it.skip(browserName === 'webkit' && parseInt(os.release(), 10) < 20, 'WebKit for macOS 10.15 is frozen.');
+  it.skip(browserName === 'webkit' && platform === 'darwin', 'WebKit for macOS 10.15 is frozen.');
+  it.skip(browserName === 'webkit' && platform === 'darwin' && parseInt(os.release(), 10) === 22, 'Modernizr uses WebGL which is not available in macOS-13 - https://bugs.webkit.org/show_bug.cgi?id=278277');
   const context = await browser.newContext({
     deviceScaleFactor: 2
   });
@@ -81,7 +82,8 @@ it('safari-14-1', async ({ browser, browserName, platform, server, headless, isM
 
 it('mobile-safari-14-1', async ({ playwright, browser, browserName, platform, isMac, server, headless }) => {
   it.skip(browserName !== 'webkit');
-  it.skip(browserName === 'webkit' && parseInt(os.release(), 10) < 20, 'WebKit for macOS 10.15 is frozen.');
+  it.skip(browserName === 'webkit' && platform === 'darwin' && parseInt(os.release(), 10) < 20, 'WebKit for macOS 10.15 is frozen.');
+  it.skip(browserName === 'webkit' && platform === 'darwin' && parseInt(os.release(), 10) === 22, 'Modernizr uses WebGL which is not available in macOS-13 - https://bugs.webkit.org/show_bug.cgi?id=278277');
   const iPhone = playwright.devices['iPhone 12'];
   const context = await browser.newContext(iPhone);
   const { actual, expected } = await checkFeatures('mobile-safari-14-1', context, server);
diff --git a/tests/page/page-goto.spec.ts b/tests/page/page-goto.spec.ts
index a613a7e036fba..79107c0d215b8 100644
--- a/tests/page/page-goto.spec.ts
+++ b/tests/page/page-goto.spec.ts
@@ -181,7 +181,8 @@ it('should work with Cross-Origin-Opener-Policy after redirect', async ({ page,
 
 it('should properly cancel Cross-Origin-Opener-Policy navigation', {
   annotation: { type: 'issue', description: 'https://github.com/microsoft/playwright/issues/32107' },
-}, async ({ page, server }) => {
+}, async ({ page, server, browserName, isLinux }) => {
+  it.fixme(browserName === 'webkit' && isLinux, 'Started failing after https://commits.webkit.org/281488@main');
   server.setRoute('/empty.html', (req, res) => {
     res.setHeader('Cross-Origin-Opener-Policy', 'same-origin');
     res.end();
diff --git a/tests/page/page-screenshot.spec.ts b/tests/page/page-screenshot.spec.ts
index 7f374ac59be73..8d0e68e12df37 100644
--- a/tests/page/page-screenshot.spec.ts
+++ b/tests/page/page-screenshot.spec.ts
@@ -323,6 +323,7 @@ it.describe('page screenshot', () => {
   it('should work for webgl', async ({ page, server, browserName, platform }) => {
     it.fixme(browserName === 'firefox');
     it.fixme(browserName === 'chromium' && platform === 'darwin' && os.arch() === 'arm64', 'SwiftShader is not available on macOS-arm64 - https://github.com/microsoft/playwright/issues/28216');
+    it.skip(browserName === 'webkit' && platform === 'darwin' && parseInt(os.release(), 10) === 22, 'WebGL is not available in macOS-13 - https://bugs.webkit.org/show_bug.cgi?id=278277');
 
     await page.setViewportSize({ width: 640, height: 480 });
     await page.goto(server.PREFIX + '/screenshots/webgl.html');