When referring to the garden or school grounds, we often refer to it as a campus. You can also use "school grounds". Also, you can use the word "stroll" in place of walk.
To stroll = walk in a leisurely way, saunter, amble, wander, meander, ramble, dawdle, promenade, walk, go for a walk, take a walk, roam, traipse,
"I strolled around the city"
Campus
= Campus refers to the area attached to the educational establishment which may include sports and training grounds, pavements and roads, buildings for education or accommodation, parkland and flower gardens.
散歩する＝楽しんで歩く
同義語
saunter, amble, wander, meander, ramble, dawdle, promenade, walk, go for a walk, take a walk, roam, traipse,
"I strolled around the city"
街をブラッと歩いた。
キャンパスは、スポーツ、トレーニンググラウンド、歩道や道路、校舎、寮、公園や花が咲いた庭がある教育機関がある場所のことです。
Since the question mainly concerns a word that can replace "the grounds of a college or university", the right word would be 'campus'. The noun 'campus' means the grounds and the buildings of a college or a university. If a student lives at the university where he/she is studying, you can say he/she lives on the campus. However, in this case, you walked through the campus. Grammatically, the word 'through' has been used as a preposition here. If you took a walk through the campus, it means that you entered the campus on one side and came out on the other side but not around it.
So, you may say:
I took a walk through my campus.
大学の敷地内ということについてなので、適した言葉はcampus（キャンパス）と言う言葉でしょう。
このキャンパスという単語は、大学の土地や建物のことを意味します。
所属する大学に住んでいる学生は、he/she lives on the campus.（彼または彼女はキャンパスに住んでいる）と言うことができます。
しかしながら、この質問の場合は、you walked through the campus.（あなたはキャンパスを通り抜けた。）と言います。
文法的には、throughという言葉は前置詞として使っています。
この例文の意味は、大学の敷地内を通り抜けたという意味で、中を回ったということではありません。
そのため、このように言うと良いでしょう。
例文
I took a walk through my campus.
（私はキャンパスを通り抜けた。）
Example sentences-
1. I live on campus and frequently take strolls around at night, it is very safe.
2. The university grounds are landscaped very beautifully, I try to walk around every day.
【例文】
1. I live on campus and frequently take strolls around at night, it is very safe.
（私は大学の敷地内に住んでいて、よく夜に散歩をします、とても安全です）
2. The university grounds are landscaped very beautifully, I try to walk around every day.
（大学敷地内はとてもきれいに庭が整備されていて、毎日歩くようにしています）
I took a turn about the campus:
To take a turn about the campus means to take a walk
around the area or the buildings of the college /university.
I strolled around the campus:
To stroll means to walk in a relaxed manner. A campus
is the grounds of a university / college.
I took a turn about the campus.
（大学の敷地を散歩しました）
"take a turn about the campus" は、大学の敷地や建物を歩いて回ることを表します。
I strolled around the campus.
（大学の敷地を散策しました）
"stroll"（散策する）は、リラックスして散歩するという意味です。
"campus"とは大学の敷地の事を言います。
The property of a college is most commonly referred to as "campus". You can also say, "college grounds" or "campus grounds".
Examples:
I took a walk around campus last night to clear my head.
I walked around the college grounds for exercise this morning.
Yesterday, I took a walk through the campus grounds.
Clear my head = stop worrying/thinking about something
大学の敷地を表す最も一般的な表現は "campus"（キャンパス）です。
また "college grounds" や "campus grounds"と言う事も出来ます。
【例文】
I took a walk around campus last night to clear my head.
（頭をすっきりさせるために昨夜大学の敷地内を散歩しました）
I walked around the college grounds for exercise this morning.
（エクササイズするために、今朝大学の敷地を歩きました）
Yesterday, I took a walk through the campus grounds.
（昨日、大学の敷地を散歩しました）
Clear my head = 何かを心配する/考える事を止める事です。
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