Here's what summarize() is expecting to get:
Name-value pairs of summary functions. The name will be the name of the variable in the result. The value should be an expression that returns a single value like min(x), n(), or sum(is.na(y)).
summarize() feeds each group of rows (identified by group_by) into whichever aggregation functions you pass to it. Then it collapses your dataframe so that there is one row for each group, with only your grouping columns plus columns for the results of the aggregation functions.
The n() function automatically counts observations in each group, so you don't need to directly process URL in order to get a count of URLs per group. Funnily enough, it doesn't matter what columns you have other than Date, the code is the same:
example_data_summary <- df %>%
group_by(year_week = floor_date(Date, "1 week")) %>%
summarize(count = n())
One note: originally, you wanted your dates formatted like YYYY-WW where WW is the week of the year. This code rounds your dates down to the date corresponding to the start of the nearest full week, but on its own it doesn't format the dates. This is a very good thing for plotting, because it means you have real date values from which to construct an axis (if you formatted the dates in your dataframe, they'd be character values and ggplot() would treat them as categories — not what you want!). If necessary, you can choose whatever formatting you need later as part of the plotting code (but ggplot() tends to make pretty good automatic guesses).
There are different standards in different parts of the world for how to count weeks. The floor_date() default is to start the week on Sunday. If you need weeks to start on a different day, you'll have to set that using the week_start parameter. For instance, to have weeks start on Monday:
floor_date(Date, "1 week", week_start = 1)