diff --git a/lib/web/cache/cache.js b/lib/web/cache/cache.js
index 45c6fec3467..5bfd94b3281 100644
--- a/lib/web/cache/cache.js
+++ b/lib/web/cache/cache.js
@@ -334,7 +334,7 @@ class Cache {
       const reader = stream.getReader()
 
       // 11.3
-      readAllBytes(reader).then(bodyReadPromise.resolve, bodyReadPromise.reject)
+      readAllBytes(reader, bodyReadPromise.resolve, bodyReadPromise.reject)
     } else {
       bodyReadPromise.resolve(undefined)
     }
diff --git a/lib/web/fetch/util.js b/lib/web/fetch/util.js
index 28d54fe946c..f53991c9440 100644
--- a/lib/web/fetch/util.js
+++ b/lib/web/fetch/util.js
@@ -1029,7 +1029,7 @@ function iteratorMixin (name, object, kInternalIterator, keyIndex = 0, valueInde
 /**
  * @see https://fetch.spec.whatwg.org/#body-fully-read
  */
-async function fullyReadBody (body, processBody, processBodyError) {
+function fullyReadBody (body, processBody, processBodyError) {
   // 1. If taskDestination is null, then set taskDestination to
   //    the result of starting a new parallel queue.
 
@@ -1054,11 +1054,7 @@ async function fullyReadBody (body, processBody, processBodyError) {
   }
 
   // 5. Read all bytes from reader, given successSteps and errorSteps.
-  try {
-    successSteps(await readAllBytes(reader))
-  } catch (e) {
-    errorSteps(e)
-  }
+  readAllBytes(reader, successSteps, errorSteps)
 }
 
 /**
@@ -1096,30 +1092,39 @@ function isomorphicEncode (input) {
  * @see https://streams.spec.whatwg.org/#readablestreamdefaultreader-read-all-bytes
  * @see https://streams.spec.whatwg.org/#read-loop
  * @param {ReadableStreamDefaultReader} reader
+ * @param {(bytes: Uint8Array) => void} successSteps
+ * @param {(error: Error) => void} failureSteps
  */
-async function readAllBytes (reader) {
+async function readAllBytes (reader, successSteps, failureSteps) {
   const bytes = []
   let byteLength = 0
 
-  while (true) {
-    const { done, value: chunk } = await reader.read()
+  try {
+    do {
+      const { done, value: chunk } = await reader.read()
 
-    if (done) {
-      // 1. Call successSteps with bytes.
-      return Buffer.concat(bytes, byteLength)
-    }
+      if (done) {
+        // 1. Call successSteps with bytes.
+        successSteps(Buffer.concat(bytes, byteLength))
+        return
+      }
 
-    // 1. If chunk is not a Uint8Array object, call failureSteps
-    //    with a TypeError and abort these steps.
-    if (!isUint8Array(chunk)) {
-      throw new TypeError('Received non-Uint8Array chunk')
-    }
+      // 1. If chunk is not a Uint8Array object, call failureSteps
+      //    with a TypeError and abort these steps.
+      if (!isUint8Array(chunk)) {
+        failureSteps(TypeError('Received non-Uint8Array chunk'))
+        return
+      }
 
-    // 2. Append the bytes represented by chunk to bytes.
-    bytes.push(chunk)
-    byteLength += chunk.length
+      // 2. Append the bytes represented by chunk to bytes.
+      bytes.push(chunk)
+      byteLength += chunk.length
 
     // 3. Read-loop given reader, bytes, successSteps, and failureSteps.
+    } while (true)
+  } catch (e) {
+    // 1. Call failureSteps with e.
+    failureSteps(e)
   }
 }