How to run tasks concurrently in asyncio?

Your crawl method is waiting after each individual task; you should change it to this:

async def crawl(self):
    tasks = []
    while self.active_fetches < self.MAX_ACTIVE_FETCHES:
        if self.urlq:
            url = self.urlq.pop()
            tasks.append(asyncio.create_task(self.fetch(url)))
    await asyncio.gather(*tasks)

EDIT: Here's a cleaner version with comments that fetches and processes all at the same time, while preserving the basic ability to put a cap on the maximum number of fetchers.

import asyncio

class Crawler:

    def __init__(self, urls, max_workers=2):
        self.urls = urls
        # create a queue that only allows a maximum of two items
        self.fetching = asyncio.Queue()
        self.max_workers = max_workers

    async def crawl(self):
        # DON'T await here; start consuming things out of the queue, and
        # meanwhile execution of this function continues. We'll start two
        # coroutines for fetching and two coroutines for processing.
        all_the_coros = asyncio.gather(
            *[self._worker(i) for i in range(self.max_workers)])

        # place all URLs on the queue
        for url in self.urls:
            await self.fetching.put(url)

        # now put a bunch of `None`'s in the queue as signals to the workers
        # that there are no more items in the queue.
        for _ in range(self.max_workers):
            await self.fetching.put(None)

        # now make sure everything is done
        await all_the_coros

    async def _worker(self, i):
        while True:
            url = await self.fetching.get()
            if url is None:
                # this coroutine is done; simply return to exit
                return

            print(f'Fetch worker {i} is fetching a URL: {url}')
            page = await self.fetch(url)
            self.process(page)

    async def fetch(self, url):
        print("Fetching URL: " + url);
        await asyncio.sleep(2)
        return f"the contents of {url}"

    def process(self, page):
        print("processed page: " + page)


# main loop
c = Crawler(['http://www.google.com', 'http://www.yahoo.com', 
             'http://www.cnn.com', 'http://www.gamespot.com', 
             'http://www.facebook.com', 'http://www.evergreen.edu'])
asyncio.run(c.crawl())

You can make htmlq an asyncio.Queue(), and change htmlq.append to htmlq.push. Then your main can be async, like this:

async def main():
    c = Crawler()
    asyncio.create_task(c.crawl())
    while True:
        page = await c.htmlq.get()
        if page is None:
            break
        c.process(page)

Your top-level code boils down to a call to asyncio.run(main()).

Once you are done with crawling, crawl() can enqueue None to notify the main coroutine that the work is done.