Your question is more related to the taxonomy of Islamic knowledge coupled with literary devices in the Arabic language (ta'mīm and takhsīs).
The Qur'an used the three words interchangeably. For example, in the following verse, sharī'ah is used to refer to the ordained ways of the religion (dīn):
ثُمَّ جَعَلْنَاكَ عَلَىٰ شَرِيعَةٍ مِّنَ الْأَمْرِ فَاتَّبِعْهَا وَلَا تَتَّبِعْ أَهْوَاءَ الَّذِينَ لَا يَعْلَمُونَ
Then We put you, [O Muhammad], on an ordained way concerning the matter [of religion]; so follow it and do not follow the inclinations of those who do not know.
— Surat Al-Jathiyah 45:18
Ibn Kathir explained sharī'ah as what has been revealed to the Prophet ﷺ from Allah through the Qur'an. This includes law, rituals, creed, history, practices, inheritance, etc. Al-Qurtubi explained that sharī'ah in this verse covers madhhab and minhāj. Al-Qurtubi elaborated that Ibn Zaid said it meant religion, as did Ibn al-'Arabi and Ibn 'Abbas; while Qatadah said it is particular to what is permissible, what is not permissible, and inheritance laws (i.e., fiqh).
It was also used to refer to religion and jurisprudence:
شَرَعَ لَكُم مِّنَ الدِّينِ مَا وَصَّىٰ بِهِ نُوحًا وَالَّذِي أَوْحَيْنَا إِلَيْكَ وَمَا وَصَّيْنَا بِهِ إِبْرَاهِيمَ وَمُوسَىٰ وَعِيسَىٰ ۖ أَنْ أَقِيمُوا الدِّينَ وَلَا تَتَفَرَّقُوا فِيهِ ۚ كَبُرَ عَلَى الْمُشْرِكِينَ مَا تَدْعُوهُمْ إِلَيْهِ ۚ اللَّهُ يَجْتَبِي إِلَيْهِ مَن يَشَاءُ وَيَهْدِي إِلَيْهِ مَن يُنِيبُ
He has ordained for you of religion what He enjoined upon Noah and that which We have revealed to you, [O Muhammad], and what We enjoined upon Abraham and Moses and Jesus - to establish the religion and not be divided therein. Difficult for those who associate others with Allah is that to which you invite them. Allah chooses for Himself whom He wills and guides to Himself whoever turns back [to Him].
— Surat Ash-Shuraa 42:13
In Al-Adyān wa al-Madhāhib 2/43-46 (Arabic only), the difference between sharī'ah and minhāj is discussed in details. Under this chapter, the topic of the synonymity of the three words dīn, sharī'ah, and fiqh was discussed. The use of the three words interchangeably is very common in Islamic literature.
In Islam, the definition of sharī'ah is:
الشريعة ما شرع الله لعباده، والظاهر المستقيم من المذاهب كالشرعة
NOTE. My own translation, so treat with care.
Sharī'ah is what Allah decrees to His servants, and what is straight and obvious from the madhāhib as in shir'ah.
Ibn 'Abbas explained that shir'ah is what is revealed in the Qur'an and minhāj is what is revealed through Sunnah.
Furthermore, it is explicitly mentioned that the word sharī'ah could be a synonym for fiqh:
كما أنه تذكر الشريعة، ويراد بها الفقه في بعض الأحيان من باب إطلاق العام ويُراد به الخاص
NOTE. My own translation, so treat with care.
And also the word sharī'ah could be mentioned to mean fiqh in some cases as a form of generalization (itlāq al-'ām) when mentioning specificities (al-khās).
As-Shatibi, in his definition of sharī'ah, considered it to be all encompassing of the entire dīn and not just fiqh:
إن معنى الشريعة أنها تحدُّ للمكلفين حدودًا في أفعالهم وأقوالهم، واعتقاداتهم، وهو جملة ما تضمنته، ومعنى هذا أن الشريعة مرادفة للدين، وليس يُراد بها الفقه وحده؛ لأن الفقه لا يتعرض للاعتقادات كما هو معلوم
NOTE. My own translation, so treat with care.
The meaning of sharī'ah is what defines to those who are required to comply the rules and regulations (hudūd) of what is said, done, and believed, i.e., it is overall inclusive. This means that sharī'ah is a synonym of religion, and it is not to be restricted to jurisprudence only since jurisprudence does not discuss creed as it is known.
This answer is not meant to address all aspects of the interchangeably-used words since this is a very wide topic to cover in one answer. I am almost certain there will be follow-up questions. I just want to summarize when one gets a bachelor's degree in sharī'ah from an Islamic university, the sharī'ah courses cover:
- Qur'an: verses and tafsīr
- Hadith books, terminology, and methodology
- Fiqh schools and principles
- Creed principles and epistles
- Islamic practices (family)
- History (sīra and tarājim)
- Farā'id (laws of inheritance)
The books studied are almost always the same across universities. The above is also almost always the taxonomy used in the classification of Islamic topics and books.