You have to fine tune the regular expression (i.e. "\\.(html|php)$") to fit your specific application.
Regular Expressions are common to many programming languages and not specific to R, if you are going to be cleaning text often, you might benefit from learning about them.
You are mixing tidyverse and base R syntax, you have to choose one, following my previous example the code would be like this.
library(tidyverse)
# Read the csv file into memory
oldata3 <- read.csv("C:/Users/FBDA17-031/Documents/OlyDash/Oldata3.csv")
# Create the new column
oldata3 <- oldata3 %>%
mutate(Games = str_remove(URL, "\.(html|php)$"))
# Write back the data frame to a csv file
write.csv(oldata3, "C:/Users/FBDA17-031/Documents/OlyDash/Oldata3.csv")